Canon doesn't indicate that Dolores Umbridge had the locket Horcrux prior to or during Order of the Phoenix. After she acquired it, she was able to produce a Patronus without any trouble while wearing the locket; it fostered her abilities, though, rather than negatively affecting them. J.K. Rowling addresses this:
Question: How did Umbridge manage to conjure a Patronus while wearing the locket when Harry wasn't able to?
JK Rowling: Because she is a very nasty piece of work. She has an affinity for this horrible object, which would help rather than hinder her.
MUGGLENET - BLOOMSBURY POST-DEATHLY HALLOWS WEBCHAT - JULY 2007
Regarding the specific dates of the locket's theft by Mundungus Fletcher and Dolores Umbridge's acquisition of it, I'm sorry but there is no TL;DR version. This is very long; the OP requested details.
According to the HP LEXICON, Harry arrived at No. 12 Grimmauld Place for the first time on August 6, 1995, under the watch of the Advance Guard. He meets Mundungus Fletcher for the first time at No. 12 Grimmauld Place; the topic of Dung having abandoned his post to guard Harry to see after some stolen cauldrons is referenced several times in chapter 5, The Order of the Phoenix. Dung and Sirius have the following conversation:
‘Sirius,’ said Mundungus, who did not appear to have paid any attention to the conversation, but had been minutely examining an empty goblet. ‘This solid silver, mate?’
‘Yes,’ said Sirius, surveying it with distaste. ‘Finest fifteenth century goblin-wrought silver, embossed with the Black family crest.’
‘That’d come orf, though,’ muttered Mundungus, polishing it with his cuff.
Order of the Phoenix - page 84 - UK Hardcover - chapter 5, The Order of the Phoenix
As it is one of these goblets that Harry later catches Dung with in Hogsmeade during Half-Blood Prince, we know that Dung did not steal the goblets -- or the locket -- prior to August 6, 1995.
On August 7, 1995, Dung attempts to stash the stolen cauldrons at No. 12 Grimmauld Place as the Order is busy cleaning their headquarters. It's during this clean-up that the locket is found.
There was a musical box that emitted a faintly sinister, tinkling tune when wound, and they all found themselves becoming curiously weak and sleepy, until Ginny had the sense to slam the lid shut; a heavy locket that none of them could open; a number of ancient seals; and, in a dusty box, an Order of Merlin, First Class, that had been awarded to Sirius’s grandfather for ‘services to the Ministry’.
Order of the Phoenix - page 108 - UK Hardcover - chapter 6, The Noble and Most Ancient House of Black
I think it can be concluded that Dung did not steal the locket prior to August 7, 1995.
Between August 7 and August 31, 1995 while continuing the cleaning on No. 12 Grimmauld Place, Dung may have stolen some items which may have included the locket, but my sense is the items were nicked by Kreacher around this time, rather than Dung.
Mrs Weasley was in such a good mood she did not even complain that they had brought Mundungus with them; he
was wearing a long overcoat that seemed oddly lumpy in unlikely places and declined the offer to remove it and put it with Moody’s travelling cloak.
Order of the Phoenix - page 158 - UK Hardcover - chapter 9, The Woes of Mrs Weasley
To me, this reads as Mundungus entering the house with his overcoat already stuffed full of items, though. I can see an argument for both points of view, for it also doesn't say, when noting Dung came along and had a lumpy overcoat, how much time had elapsed. So, could he have taken the goods right away? Perhaps. I tend to think not, though.
So I'm going to wager the locket was nicked by Kreacher and stashed away sometime between August 7, 1995, and August 31, 1995 (the dates the Order and the kids were at Grimmauld presumably; the kids would have returned to Hogwarts on September 1, 1995)
As well, Sirius was living in Grimmauld 24-7 for the entire duration of Order of the Phoenix, which would have been a major deterrent to Dung coming in and cleaning the place out. Sirius is a formidable wizard and had spent twelve years in Azkaban -- if anything heightens the senses and paranoia and causes hypervigilance, it would likely be Azkaban. As well, Buckbeak was living in Grimmauld.
Once Harry returned to Hogwarts, the Order kept watch over Harry, even though he was at school.
Mundungus spied on Harry, Ron, and Hermione on October 5, 1995, in the Hog's Head, as they found the DA; Sirius tells Harry about this shortly thereafter. No mention is made of stolen Black belongings by Sirius, but to be fair, Sirius might not have cared that his property was taken. He was prepared to throw all the Black heirlooms away during the summer, including family pictures.
Fast-forward to June 1996 and the Battle of the Department of Mysteries where Sirius is killed. It's possible Dung stole the Black items immediately following Sirius's death; later on down the timeline, in Half-Blood Prince, Harry accuses Dung of doing just that on October 5, 1996 during a Hogsmeade visit.
‘You took that from Sirius’s house,’ said Harry, who was almost nose-to-nose with Mundungus and was breathing in
an unpleasant smell of old tobacco and spirits. ‘That had the Black family crest on it.’
‘I – no – what –?’ spluttered Mundungus, who was turning slowly purple.
‘What did you do, go back the night he died and strip the place?’ snarled Harry.
‘I – no –’
Half-Blood Prince - page 235 - UK Hardcover - chapter 12, Silver and Opals
Dung denied stealing the Black family goblets the night Sirius died; on the other hand, he tried to deny taking them at all, which clearly wasn't true. I think it's possible Dung could have taken the goblets and locket as early as June 1996, immediately following Sirius's death. That would have been an exceptionally low thing to do, even for a sneakthief like Dung. Nevertheless, I think this is the earliest that Dung might have stolen the items, based on canon references. When Harry confronts Dung in Hogsmeade it is October 5, 1996, and Dung by that time has the Black heirloom goblets. I would say presumably the locket had also been taken by this time, as the locket and the goblets had been put in the same trash bag for disposal.
Logically, Dung likely stole the locket between June 1996 and October 1996 -- this is what I've concluded as being most reasonable a guess. However, I acknowledge there is a possibility Dung did not nick the locket until later. Yes, Dung could've accessed Grimmauld between October 1996 and June 1997, up until Dumbledore's death. Although Dung was a member of the Order of the Phoenix and had likely been told the location of the Order's headquarters, Mad-Eye Moody very well could have put up additional protective spells and charms around the house that might have kept Dung out following Dumbledore's death. Maybe Dung could've gotten in; maybe not. We don't know. June 1997, up until Dumbledore's death, is probably the latest Dung could have stolen the locket.
On August 2, 1997, Kreacher tells Harry, Ron, and Hermione that Dung has stolen the locket.
‘Mundungus Fletcher,’ croaked the elf, his eyes still tight shut. ‘Mundungus Fletcher stole it all: Miss Bella and Miss Cissy’s pictures, my mistress’s gloves, the Order of Merlin, First Class, the goblets with the family crest, and, and –’ [...] ‘– and the locket, Master Regulus’s locket, Kreacher did wrong, Kreacher failed in his orders!’
Deathly Hallows - page 158 - UK Hardcover - chapter , Kreacher's Tale
As mentioned, Kreacher tells his tale about the locket on August 2, 1997, and Harry orders Kreacher to find Mundungus and bring Dung back to Grimmauld. Kreacher is gone for three days. So he returns on August 5, 1997 (chapter 11, The Bribe in Deathly Hallows), with Dung, and Dung tells the trio his story of how Dolores Umbridge took the locket in lieu of arresting him for peddling without a license. So we know as of August 5, 1997, Dung does not have the locket anymore.
On September 2, 1997, Harry, Ron, and Hermione infiltrate the Ministry of Magic and steal the locket Horcrux back from Dolores Umbridge.
By my calculations, Dolores Umbridge may have come into possession of the locket Horcrux by as early as June 1996 through as late as August 4, 1997. This allows for the possibility of Dung having stolen the locket immediately following Sirius's death. However, October 5, 1996 is when we see Dung in possession of the stolen Black heirloom goblets when Harry catches up with him in Hogsmeade. Since the locket and goblets were together in Order of the Phoenix, I think Dung probably stole the items together between September 1, 1996 through October 4-5, 1996. For this reason, I think it's more likely that Umbridge snagged the locket from Dung sometime between October 5, 1996 and August 4, 1997. This is a ten month window, quite a lot of which is during winter and spring. What I deduce as direct canon evidence stops here.
However, I'd like to throw out a couple more ideas that are definitely within the spirit of canon. Regarding the sale of the locket, I believe, if given a choice, a person would likely be less inclined to stand out on the street and peddle their wares during the very coldest months, which is November through March in the UK. This is just a theory! I'll postulate that it's possible there is a lower likelihood of Dung having been out on the streets selling his stuff in Diagon Alley during the coldest months. Were this to be true, it knocks out November and December of 1996 and January, February, and March of 1997. This leaves 26 days in October 1996, 4 days in August 1997, and April, May, June, and July of 1997 that Umbridge could have nicked the locket from Mundungus, which reduces Umbridge's acquisition window to five months and four days.
To attempt to narrow it even further, Dumbledore was killed in the latter part of June 1997 (see the first two pages of chapter 30, The White Tomb in Half-Blood Prince); Umbridge attends his funeral in late June. No mention is made of her wearing a locket, nor is anything else she wore described, except for her velvet hair bow (chapter 30, The White Tomb in Half-Blood Prince) So it's possible Umbridge didn't own the locket by June of 1997, or perhaps she did have it and just didn't wear it to Dumbledore's funeral. If it was that she didn't have the locket at the time of Dumbledore's funeral, that eliminates October 1996, and April, May, and the majority of June 1997 as possible acquisition dates. It would put the time of her procuring the locket during either the last few days of June, all of July, or August 1-4, 1997, which is a window of approximately six weeks.
SHORT VERSION (CANON): Umbridge took the locket between either June of 1996 (Sirius's death) or October 5, 1996 (Harry confronts Dung in Hogsmeade) and August 4, 1997.
MEDIUM VERSION: Umbridge took the locket between June of 1996 and November 1996 or between April 1997 and August 4, 1997, if one concludes Dung might have been less likely to peddle during the coldest months.
LONG VERSION: Umbridge took the locket between either October 5, 1996 (Harry confronts Dung) and October 31, 1996 (November begins the coldest months), or between the latter part of June 1997 (Dumbledore's death) and August 4, 1997 (the day the trio interrogates Dung at Grimmauld), if one accepts Dung didn't steal the locket until after August 31, 1995 (after the Order presumably cleared out from Grimmauld), that Umbridge didn't have the locket at the time of Dumbledore's death, and that Umbridge didn't nick it from Dung until after the latter part of June 1997 up to August 4, 1997. Also note the Death Eaters were running the Ministry behind the scenes as Half-Blood Prince turns over to Deathly Hallows. Umbridge, as a Death Eater sympathizer/supporter, may have felt it was safe to steal from a peddler with her side now in charge of law enforcement (or lack thereof!)
I have a hard time with math -- please leave a comment if I've made an obvious and egregious calculation error. :)