When Darth Vader and Princess Leia meet on board the Tantive IV in Star Wars Episode IV: A New Hope for the first time, does Darth Vader know she's his daughter?
At that point he doesn't even know Luke is his son and that's what starts his interest in finding "Young Skywalker."
He shows his surprise at learning that Luke has a sister in Return of the Jedi when he says he sees Luke's feelings are strong for his friends – then adds, in surprise, "Sister."
Wookieepedia states (when describing the duel between Luke and Vader on the new Death Star):
Vader, who had been probing Luke's thoughts, discovered that besides Luke he also had a daughter: Leia Organa
However, no attribution for this assertion is given in the article.
To re-use part of my answer to an unrelated question Why do we hear Leia's Theme during Kenobi's death?:
At the time of working of "A New Hope" (e.g. prior to 1977) - it was NOT known to anyone (including Lucas) that Luke and Leia are family. Since Luke was Anakin Skywalker's son from the beginning of ANH, that means Leia wasn't meant to be his daughter, until at least mid-work on Episode V.
To be more specific, even an early Leigh Brackett's 1978 draft of "The Empire Strikes Back" didn't have that relationship established yet - it was introduced later (source).
Another related evidence for Luke and Leia not yet being related was 1978 pre-ESB book that was a sequel to "ANH" - Alan Dean Foster's "Splinter of the Mind's Eye". It develops the theme of romantic interest between the two (presumably, in part, since Foster and Lucas didn't know if ANH would be a big hit and if Harrison Ford would be interested in the sequel should the sequel be based on "Splinter")
To further reinforce the answer "no", spoilers below:
Dark Lord: The Rise of Darth Vader has some info on how Darth Vader went to Alderaan to "inquire" Bail Organa about anti-Empire protests taking place outside the royal palace on Aldera. Breha Organa narrowly avoids meeting Vader by infant Leia's "I'm hungry" shrieks. If he knew anything, he would march into the royal chambers and forcefully take the baby, I guess.
I honestly think when Sidious told Vader he'd killed Padme, Vader's and Sidious's interpretation was mother and child both died. Why Padme never had a pre-natal scan is puzzling?? Anyway,Luke only really came to light after he blew up the death star. With Obi-Wan also being there, Vader also mentioned when trying to kill Luke on the death star trench run. "the force is strong in this one".
Vader also told Grand Moff Tarvin "her resistance against the mind probe was considerable". But didn't realise even then she was his daughter. Even Lucas didn't know what was happening or he wouldn't have had Luke & Leia smooching in ESB. But Leia & Luke had a force connection on Cloud City when he reaches out to her for help to come and get him.
Leia did not have to suppress her force signature any more than the seperation of the twins across the galaxy already suppressed their force signature. Because she was raised on Alderaan under royalty, the nature and setting of her upbringing basically did not allow the force to be easily recognized because it was never developed. Obi Wan Kenobi remained on Tatooine keeping an eye and waiting for Luke to come of age. It was because Obi Wan already obviously knew the importance of Luke's bloodline that he was brought into the war. Vader was under the impression that he was only to have one child, and when Sidious rescued Anakin after his defeat on Mustafar by Obi Wan, Sidious told Vader that he was responsible for Padmes death. There was the assumption the child was lost as well.
I think leia was using the force in a new hope without even realizing it, she was resistant to the mind probe. But because vader was so deep in the darkside and had no struggling conflicts a small sign of the force isnt significant enough. He believed his kid(s) dead that's what he was told so he would have no reason to suspect anything. Not until empire anyway
How could Vader not know?
My answer is that he did.
They interrogated Leia on the Death Star. They used pharmaceuticals on her. I'm sure they took something resembling a DNA scan. They had a clone army, so they were aware of DNA. If they had, and it tracked to Vader, they would have headed straight to the Emperor. Why couldn't Vader have tried to pull the information out of her mind, As Kylo Ren attempted to do with Rey in TFA. In Cloud City when they freeze Han, and she stares at Vader, she was enraged - and force sensitive - that should have created a disturbance in the Dark Side, which Vader would definitely have noticed.