That seems to be so, under New York City Administrative Code § 10-303 (at least with some narrow exceptions), as interpreted by the New York Police Department. Does anyone whether there are other rules that might keep this from being the case?
Query: If the Second Amendment is incorporated against the states, can this be constitutional? As I noted with regard to a much less restrictive Illinois limit on 18-to-20-year-olds, the age of majority has indeed historically been 21 until the early 1970s, though now it's 18 virtually everywhere, including in New York.
Would the right to keep and bear arms not fully apply to under-21-year-olds, the way some constitutional rights today don't fully apply to under-18-year-olds (consider the right to sexual autonomy, the right to marry, the right to abortion, which could be limited through certain kinds of parental consent laws, and likely the right to bear arms itself)? Or does the right apply to all adult citizens — unless otherwise disqualified by reason of felony conviction or the like — under today's age of majority, regardless of what the age of majority was at the time? Or has the right always extended to everyone 18 and above, regardless of the age of majority for other purposes?
These questions would also have some importance in other states that allow long gun possession for 18-to-20-year-olds but ban handgun possession until age 21, and also as to the federal government, which makes handguns harder for under-21-year-olds to get.