Chris's "real" age was listed (29), so it isn't like she refused to give her age out (although it may be fake). We don't know how old Sharon is supposed to be, but that doesn't mean she refused to give it out. In order for me to reply to this, I need to know who "them" and "they" are here. Vermouth's audience matters to how I explain this. The need to disguise may be the reason she went into acting in the first place, but that doesn't mean she is an expert to begin with. As for becoming Toichi's apprentice, Vermouth could want to improve on her disguise abilities for work, but it is also important to consider that the older Sharon gets, the more involved the aging makeup will need to be, assuming Vermouth is not aging and is maintaining a 20-30 year old face. As long as Sharon is supposed to look 40s or younger, she will simply need to maintain minor makeup to add superficial streaks and lines, made even simpler as she is an actress who is expected to look younger than her current age. That said, once Sharon reaches the age of ~50, her makeup routine is going to need to get a lot more advanced to account for the expected effects of aging which start to accelerate. We can ballpark Sharon's would be current age somewhere between 45 and 65, with 50-60 most likely, based on how she looks and Gosho's common childbearing age range. That means twenty years ago, when Sharon started with Toichi, she is anticipating the approaching age where her makeup would be expected to get a lot more involved, and she is making sure she has a few years under her belt with Toichi so she is advanced enough by the time she starts needing it.
We have no idea if whatever Vermouth took (but this applies to APTX 4869 as well) works by subtracting ten years from someone's current age, vs say, turning a person into prepubescent child no matter their starting age, or affecting different people differently. Keeping all this in mind, we can now construct a timeline for Vermouth...
~probably 30-35 years ago - Sharon debuts on screen, her parents die in a fire (Sharon's age ≈ 15-25),
30 years ago - Sharon pretends to be pregnant with Chris (Sharon's age ≈ 20-30), she wins an Oscar, her husband dies.
30 years ago - the Miyanos are called in to research
~20 years ago - Sharon started with Toichi (age ≈ 30-40)
~20 years ago - Vermouth killed Jodie's father with a 20-30 year old face
~10 years ago - Chris is old enough to switch into as a role, Chris goes independent from Sharon (age = 19)
< 1 year ago - Sharon dies (age ≈ 50-60)
< 1 year ago - Chris comes out into public besides movies and publicity interviews. (age = 28)
Looking at that timeline, something unusual should stand out. The Miyanos came in around the exact time Sharon faked being pregnant (29 years + 9 months). If one tries to take into account the time needed for the Miyanos to build trust in the organization and to make a breakthrough in their anti-aging serum, Vermouth's aging problem seems to have occurred before the Miyanos even get there. This means whatever happened to Vermouth was not related to Atsushi and Elena's APTX 4869. Instead, it is more likely the inverse situation occurred: Elena and Atsushi were called in to research Vermouth.
We also need to consider what happened to Sharon when she took whatever it was that halted her aging. Lets assume that it took some time, lets say 5 years, for Vermouth to figure out she was not aging and for the Org to find the Miyanos and recruit them in reaction to this.
If we assume a type of drug that doesn't cause any deaging but just stops aging, then that means we have a total time of 35 years since Vermouth's aging incident. Unless we consider a drug that makes you older and stops you aging, that means Sharon should be older than 55 at least in order to have a minimum 20 year old face.
It also means that Sharon would have had to start getting into high level show business immediately (less than 5 years after finding out) to acquire the necessary disguise skills. She would also have to kill her parents who could possibly notice the discrepancy eventually.
Personally, I have a different take on the theory. I think Vermouth's natural face looks more like she is 25-30, which would put Sharon by necessity up to 60-65, which is too old for how Vermouth is making Sharon up to look. Furthermore, the choice of using a fire to kill one's parents can be used to make the bodies and all personal effects nigh unidentifiable, given the forensics of the time. The timing of the deaths, the day Vermouth achieved high popularity with an Oscar and would put her parents under scrutiny, very much matches up with how Sharon died when Chris debuted.
I think it is possible that Vermouth is in fact Sharon's mother as well, making her about 80-85 years old, and on her third generation of disguise. She stopped aging 55-60 years before the current timeline. It works well within the timeline of what we know.