Did Emperor Palpatine know that Padme had given birth to twins while it was happening? Was it his long-term goal to have Vader only discover his offspring later, at some "correct" moment?
The canon, by direct reading and by inference, does not seem to suggest so:
Even though he knew of Anakin's relationship with Padme and the fact that she was pregnant, there is nothing to suggest that Palpatine was aware that Padme was pregnant with twins, as opposed to one child, in the first place.
I couldn't find any references to Palpatine either knowing, or using that knowledge to some purpose.
The absence of that knowledge appears to have specifically hurt Palpatine in ways that he could have reasonably avoided and should have wanted to avoid.
Vader never pursued an investigation into it (and did not seem to know anything about it at all until the end), though others did, and there is no evidence that Palpatine learned about it through those other investigations.
There is no evidence that I can find in the canon that anyone other than Obi-wan, Yoda and Bail Organa (and possibly R2-D2) knew that Amidala had twins. Her pregnancy was not a secret but she did not reveal to Anakin that she was carrying two children, if indeed she herself knew, only that she was pregnant. Therefore, unless the Emperor knew specifically about both of them then he and anyone else who was looking for them would probably only be looking for one child.
I could not find any evidence in the canon that Palpatine knew that Amidala had successfully given birth before dying, either when it happened, or afterwards. The answers in this question and this question suggest that he did not know at the time or at any time after that, based on how he behaved after Luke came out into the open. See also here and here for reference.
Answer by inference
There is also no evidence that Palpatine actually did anything about it, if indeed he even did know; there were no programs of spying or clever stratagems. In fact he suffered repeatedly precisely because he did not apparently know about this. Leia was a public figure who did not hide (most of the time) and Luke never left Tatooine after arriving there (until ANH), so keeping tabs would not have been difficult, and yet:
If Palpatine had been aware of the birth he would have been aware of Padme's location on Polis Massa. Yoda, Obi-Wan and Bail Organa were all present there and yet Palpatine shows making no effort to pursue them or investigate Polis Massa itself.
If he had known about Leia, his surveillance on Bail Organa would have been far greater including his role in the Rebel Alliance. Although Palpatine was aware of the budding Rebel Alliance through Galen Marek, and knew of Bail's anti-Imperialism (going back to the Clone Wars days) plus his involvement in the Corellian Treaty incident, he did not appear to be aware that Leia was actively involved until too late. Leia only participated in the Corellian Treaty via holotransmission and was not seen by Vader.
Although Leia was known by that time to be somewhat anti-Imperial (she had been detained earlier on Kashyyyk), her role in the Alliance was not known even though her father's role was. In addition, Leia was physically present at a later meeting on Kashyyyk to formally establish the Alliance wherein the Emperor could have found Garm Bel Iblis, Mon Motha, Rahm Kota, Leia and Bail Organa, all key members in the Rebellion that the Emperor had helped create and knew was going to be a big problem (TFU ending).
If he had known about Leia, he would have been able to prevent the theft of the Death Star plans or at least done far better in attempting to prevent it. Even though the Death Star was known to the Rebel Alliance at that time, he did not know that Leia was necessarily a part of it and too late did Palpatine/Vader attempt to stop her as discussed in my answer here.
If he had known about Luke, he would have had an opportunity to intercept the stolen Death Star plans. Palpatine presumably knew at some point that Leia had been intercepted over Tatooine, and Luke never left Tatooine after arriving there, so if Palpatine put a) the stolen plans, b) Leia and/or Luke, and c) Tatooine together, the Empire's attempts to retrieve the plans should/would have been far more effective.
If he had known about Luke, he would have been able to find at least Obi-Wan Kenobi on Tatooine, and perhaps even Yoda later on Dagobah.
Most importantly (to him), Palpatine would have been far better prepared for facing Luke when he finally did so.
I also cannot see any compelling reason for how Palpatine would use that sort of information to his benefit regarding Vader:
He had already pulled Vader more fully into the Dark Side by revealing to Vader that he had killed Padme, so revealing that he had been lying all this time about his children (though he was not necessarily lying about Padme) could not conceivably have helped him.
It is unlikely that Palpatine was planning on having Vader kill his children as a further test of loyalty or to push him even further to the Dark Side, especially since Palpatine was far more interested in replacing Vader anyway, which he tried to do with Luke, and with Galen Marek/Starkiller before him.
Although many people investigated her death after the fact, as mentioned here and here, there is no direct reference that Palpatine did so. There is an unsourced reference that Palpatine did block Senator Silya Shessaun from investigating Amidala's death, and that Vader was aware of Captain Typpho's investigations, but I could not find any more information on that, such as why he did so.
Even though Padme's body was available for inspection before her funeral (and potentially after it; she was entombed, not cremated), there is no evidence that Palpatine bothered to do so even if only to confirm what he may or may not have suspected; though this doesn't prove it either way.
There is a Legends reference that comes fairly close to Palpatine and it can be found in The Last of the Jedi: Death on Naboo by Jude Watson. The Last of the Jedi series tracks the adventures of Obi-Wan Kenobi and Ferus Olin and describes how Inquisitor Malorum investigates the death of Padme and her pregnancy, first on Polis Massa, then on Naboo.
The back-story of Malorum vis-a-vis Padme is that Malorum knew that Anakin and Darth Vader were the same person, though he told no one about it. He wanted Vader's job so he investigated him and this investigation led him to rumors about Polis Massa (where Padme had died) and from there to Naboo. On Naboo he discovered that before Padme's death, she had given birth, but believed that she had only had one child, and he was killed before he could transmit this information to anyone or use it in conjunction with Palpatine/Vader.
N.B.1. As an interesting side note, Palpatine and Leia came face to face in 1 BBY when she was presented by her father at the Imperial Palace before she replaced him as Senator. Darth Vader was there as well.
N.B.2. I am not claiming that it makes sense that Palpatine never seemed to investigate any of this, only that it appears that he did not know and did not attempt to find out. Leaving a loose string like that untied, knowing what he did about Anakin and Padme, seems like a colossally stupid thing to do, though there may be in- and out-of-universe justifications for it (he was busy at the time organising his Empire, he was extremely arrogant, he didn't want to tip-off Vader or anyone else or raise the issue again, he assumed that if the child[ren] had lived he would know about it et cetera).
How Palpatine knew of Luke is a mystery that has not been explained as if yet. However, there is evidence that the Emperor may have known prior to Episode IV.
There is a statement Kenobi states in Return of the Jedi, that is curious:
"...To protect you both from the Emperor, you were hidden from your father when you were born. The Emperor knew, as I did, if Anakin were to have offspring, they would be a threat to him. That is the reason why your sister remains safely annoymous..."
Palpatine also makes another curious statement that suggests that he has sensed Luke before in Jedi:
VADER: My son is with them.
EMPEROR (very cool): Are you sure?
VADER: I have felt him, my Master.
EMPEROR: Strange, that I have not...
Palpatine did also know of Padme's death before any news ever broke out informing anyone of this (even Vader was not sure of Padme's wellbeing), and he was the one informed Vader of Luke's existence. How did he come across the information? We do not know. We can only assume that he must have been of aware of Luke's (and by Kenobi's statement, probably also Leia's) existence.
Palpatine did know of Anakin's marriage to Padme, her pregnancy and Luke and quite possibly, Leia.
Palpatine was the one who brought Anakin and Padme together.
Anakin never told Palpatine of his marriage to Padme but Palpatine knew. How did he come across this information? Well, because he was the one who played matchmaker between them behind the scenes. The second film explains that Anakin has harbored feelings for Padme since he was a little boy in The Phantom Menace and Anakin didn't do a good job of hiding it. Attack of the Clones also shows that Palpatine and Anakin are good friends. I'm sure Anakin brought up the subject of his crush with Palpatine and Palpatine knew of his feelings for Padme. Padme was almost killed twice in the second film: at the landing platform and almost in her Coruscant apartment. Both were carried out by Zam Wessell but under the orders of Palpatine under the names of Jango Fett and Count Dooku. After the attempt on her life, Padme chats with the Jedi and Palpatine in the Chancellor's office at the beginning of the film. Palpatine suggests that Obi-Wan protect her because the "thought of losing her is unbearable." Knowing Obi-Wan's padawan was Anakin, it would force the two together. The second attempt was used to let Anakin be alone with Padme. It would only be a matter of time before something formed out of it (and we know, it did). When Palpatine reveals himself to be a Sith Lord to Anakin in Episode III, he says "Join me and you'll be able to save your wife from certain death" and Anakin's reaction is total shock ("What did you say?") This leads to the fact that Anakin did not tell Palpatine his secret nor of his nightmares. Palpatine knew of his wife because he was the one who brought them together, knowing she was Anakin's ultimate downfall. Since Anakin was The Chosen One who was to kill the Sith, Palpatine decided to turn his threat into his ally and used the person he loved most of all to bring him to the Dark Side (aka Padme).
Palpatine created Anakin's nightmares of Padme dying
Palpatine knew of Anakin's marriage to Padme and of Anakin's fear of losing those he loved to death. In the beginning scenes of Episode III: Revenge of the Sith, Palpatine reminds Anakin of "what he told him of his mother and the sandpeople." Palpatine was the only one Anakin told about this, besides Padme. Palpatine somehow knew of Padme's pregnancy (how the information was uncovered is unknown to us, the audience), however it is strongly implied that Palpatine is the source of the "visions." He brings up the matter of Plageuis being able to save those he loves the most from death. He knows death is Anakin's biggest fear. He implied the apprentice to be himself to Anakin but this is false as he later tells the newly-turned Vader later on in the film, "To cheat death to the power of only one has achieved but if we work together, I know we can discover the secret" This means he was lying to Anakin before when he said the Plageuis taught his apprentice "everything" he knew. It was a rouse to catch Anakin's interest, not to be taken as whole truth. The second imply was when he revealed himself as Darth Sidious to Anakin and he stated "Join the dark side of the force and you'll be able to save your wife from certain death." Anakin is shocked by the statement ("What did you say?") because he never told Palpatine of his nightmares nor that is was Padme who was dying in the dream. The only ones who knew was Padme and Yoda (though he didn't know Padme was the person dying). Anakin kept this oddly, to himself. This means that Palpatine had to have created the dreams to have known who was dying in the dream and her dying in childbirth means he knew of the pregnancy.
Palpatine killed Padme during Anakin's suit construction
This goes with the fan-theory that Sidious killed Padme and that she didn't die of a broken heart. I have to agree with this. There is evidence for this if you look deeper in the film storyline and plot. You know that Padme is quite alive after her encounter with Anakin on Mustafar and that the droids can't explain why she is dying but can only guess that she "lost the will to live." Padme is the only one who mentions that her heart is "breaking," so we assume she died of a broken heart as no one mentions her cause of death. Anakin sensed her during his operation and asks Sidious how she is. Sidious knows that she is dead. And how did he know that she died? The dream was that she died in childbirth, not of a broken heart. Throughout the visions, Padme is screaming for Anakin to "help her" and for some reason, she can't "hold on." This can mean that she felt Anakin and Anakin felt her throughout the entire process. When her life force vanished, Anakin sensed this and questioned what happened to his wife, "Where is Padme? Is she safe?" and later remarks, "She was alive! I felt it!" Hence, why she tells Obi-Wan there is "still good in him," and seems to struggle to hold on to consciousness. If this theory is true, Sidious would have sensed her giving birth. And of Vader? Did he sense something too? We'll get to that later. This theory makes sense, though. Sidious used this as the final breaking point to turn Anakin fully into Vader (hence his grinning when Anakin mourns Padme's death) and lies to him by telling him that he "killed" his wife.
Palpatine tells Vader of his son's existence, how did he know of Luke? Where did he get this information?
Alrighty, so we get to the next point of how the Emperor knew of Luke. In The Empire Strikes Back, Vader learns he has a son from the Emperor ("I have no doubt that this boy is the offspring of Anakin Skywalker"). How did Palpatine come across this information? Where did he learn this? It is not stated. Palpatine knew of Padme's pregnancy and if he killed her, would have sensed her in childbirth. In fact, we actually know the Emperor has sensed Luke before! In Return of the Jedi, Vader tells the Emperor that Luke is on Endor and that he sensed him there and the Emperor replies with, "Funny how I have not sensed him." This would make sense if the Emperor killed Padme and sensed the children but could not locate them. When Luke arrives on the picture, Palpatine could be able to figure out that he sensed this boy before and could piece together that the boy named Luke Skywalker was the same baby he sensed at Padme's death. What about Anakin? He sensed Padme before during his operation ("She was alive! I felt it!"). After the Emperor states his fact of Luke being Anakin's son, Vader replies with, "How is that possible?"
The Emperor answers, "Search your feelings Vader, you know it to be true." Hmmmmm.......
Obi-Wan tells Luke that he and Leia were hidden for protection. But if nobody knew of the their births, why protection? Somebody had to of known about them.
When Luke finds out the truth of his parentage, he talks to Obi-Wan in Return of the Jedi and the subject of his twin sister is brought up. Obi-Wan tells Luke, "To protect you both from the Emperor, you were hidden from your father....The Emperor knew as I did that if Anakin had any offspring, They would be a threat to him..."
The Emperor knew as Obi-Wan did? They hid the twins so that the Emperor and Vader could not find them, but it doesn't mean that they did not know of them. In the prequel trilogy, Yoda only states, "Hidden safe the children must be kept." But if the Emperor told Vader he killed his wife, it is possible that he could assume that he killed their child as well. Remember when he told Padme of his nightmare? She questioned what happened to their child, and he responded that he didn't know what happened to their baby. We can assume he sensed Luke, but not Leia.
Palpatine may have known about Leia too.
Vader did not sense Leia's birth, this is made clear in the sixth episode. The Emperor never states of Leia or Vader having a daughter. Obi-Wan states in Return of the Jedi, "To protect you both from the Emperor, you were hidden from your father when you were born. The Emperor knew as well as I that if Anakin had any offspring, they would a threat to him, that is the reason why your sister remains safely anonymous." This could mean that Vader might not have sensed a connection between himself and Leia but Palpatine might have. He chose to keep quiet about it because she would have been a threat to Vader. Palpatine never mentions Luke for a good while between the fourth and fifth films. It is possible that Leia's heritage may not have been a secret after all....Vader did not know the fate of his child in his dream and this most likely was what happened in real life. He may have sensed Luke, but not Leia.
Luke was actually going by the name Skywalker, if there had been any kind of active search for Anakin's kids by the Empire he would have grown up a Sith or never grown up. I would imagine the public story was something like Amidala died in childbirth and the kid didn't make it and would have been made public in the time between Sheev retrieving Anakin on Mustafar and him being put all the way back together so he knew Padme was dead and didn't bother worrying about a dead child. Leia was hidden by a name and a cover story, Luke was living on his dad's home planet under the name Skywalker...
No Sheev didn't know the kids survived until the middle of the original trilogy and I don't think he knew about Leia at all. Vader pulled that right out of Luke when they were in close proximity and Luke was angry.