Read the law carefully...
Maybe even shell out the bucks for a lawyer to be certain.
They have probably changed the law since I lived there 35 years ago, but back then, the New York law read that it was up to the issueing agency the minimum age requirement. I got my NY pistol permit in Saratoga county at age 18. So did my younger brother two years later. The judge agreed and signed the permit.
You couldn't buy a handgun until you were 21, but you could own (posess) one. We found out about this after an auto accident with both my Mom and Dad involved. Fortunately, their injuries were relatively minor (broken nose -Mom and 15 stitches - Dad) but had my father been killed, his handguns, and he had 6 at the time, would have had to have been turned over to the state, if there were no other permit holders in the household. Turning them in to the State Police, they would be kept a certain amount of time (30 fays, I think) and then destroyed. Turned in to the County Sherriff, they would be kept as long as needed for a permit application to be processed. Mom started her application before her nose even healed. And a few years later, as we each turned 18, we applied to have all the handguns (Mom's and Dad's) in the house listed on our permits. I heard of one case in those days where a judge issued a permit to a 14 year old, who ran a trapline up near the Canadian border.
Read all the applicable laws carefully. If even one says you must be 21 years old, then you are stuck until then. But if the relevant portion of the laws have not been changed since the mid 1970s, then it is up to the issuing authority. And if you can convince them you have a valid reason (and good luck on that) you can get one. But you still can't buy a handgun from an FFL dealer until you are 21, by Federal law.
I hope, for your sake, that the law is still as it was, but knowing the way those people work, it probably has been changed. They are really, really compulsive about handgun permits nowdays. 30 years after I moved out of the state, they sent me a letter, asking where the handguns listed on the permit were, telling me that, since I was no longer a state resident that my permit was invalid, and asking for the physical paper permit back!
Wanna guess how I responded?