Before I state what I think within the realm of DC, I'd like to give an old theory that I came up with to logically explain Vermouth's existence. As well as another which I can't validate within my mind due to certain aspects.
Sharon Vineyard was the original Vermouth. She was apart of the Black Org in secret, possibly using her influence to get certain things. She created a plan to bring up her child as the heir to become Vermouth. Which by that point became the pinnacle of mastery of disguise and deception within the Org. She had her child and named her Chris. She brought her up to be the worlds best actress, and to make her into the perfect spy for the Black Org. When she finished growing up, her mother Sharon, had her undergo surgery to make her into her exact replica of her when she was younger to make it seem as if she never ages. She continued on with her mother's legacy, with the BO Boss accepting Sharon's plan to pass on the name of Vermouth to her child. As for the fingerprints, she most likely had the exact copy of her Sharon's finger prints grafted to her skin. The only way for this theory to be confirmed is if Sharon's DNA is on file, and Chris's DNA is not a match.
(This is tied into my theory of the BO boss as well, which I'll explain after what I think within the DC universe with all of its peculiarities)
Theory #2: Sharon faked her own pregnancy and proceeded to get plastic surgery, and disguised herself as the older Sharon Vineyard. The only problem with this is that though it is possible to make yourself look younger, the degradation of the bones and body are not so easily reversed. So it would be hard for Vermouth to be as active if she really is the age that she is meant to be as one person.
Those explanations however disregard everything we know of the DC universe, and the BO secret projects. We are made to assume that the BO has several projects, which include, iirc, a drug to give youth. The likelihood of taking APTX4869 is slim, because the reduction in age is merely a known anomaly. While it is possible that Sharon took it with a premade antidote she prepared, it seems unlikely because it would be too risky. The more likely thing that happened is that she took a working prototype of another drug the Org is working on as a guinea pig to be younger again (and her dedication could be an added reason to be why she is the boss's favorite). It was successful and she regained her youth. Thus she is actually the same person, and likely under the watch of the science department to ensure the drug's success.
Coming back to the first one, and though this is OT, I'll give how this links to my theory of the BO boss...
Chris Vineyard, as the daughter of Sharon, was robbed of her childhood and became bitter that she had to undergo all of that to take over her mother's position. Before her mother's death, she was disguising herself as her mother and criminals preparing herself for the work at the BO. This is who we saw during the NYC case. She got saved by Ran and Shinichi, and though she didn't think much of it at the time, she had found the person she needed to destroy the org she had become so bitter towards. She realized that after her mother died, and started hearing that Kudo Shinichi was a great detective only in HS, that he would be the perfect instrument to bring down the Org. Unfortunately before she could do anything he was poisoned by Gin, and she thought he was dead. When she discovered that he was not dead, only shrunken it wasn't long before she realized that Edogawa Conan was the shrunken shinichi. And she realized that he was the silver bullet, because no one would suspect a kid. She started her revenge by killing the Boss, and assuming his position using disguising techniques. (whenever you see her talking to the Boss she is really just talking to herself using a relay or one of her underlings) She is biding her time until Shinichi is ready to bring down the org.