It is in New York Penal Law section 400.00.
Here's part of it-
Licenses to carry, possess, repair and dispose of firearms
1. Eligibility. No license shall be issued or renewed pursuant to this section except by the licensing officer, and then only after investigation and finding that all statements in a proper application for a license are true. No license shall be issued or renewed except for an applicant (a) twenty-one years of age or older, provided, however, that where such applicant has been honorably discharged from the United States army, navy, marine corps, air force or coast guard, or the national guard of the state of New York, no such age restriction shall apply...
Unfortunately there is a lot of variation on New York Pistol Permits from county to county. The law certainly says that you can apply for the permit before 21 if you are honorably discharged. There are other restrictions that vary from county to county and even change when the county court judge changes because of retirement or elections. They may require training classes, they may restrict permits to hunting and target shooting only. Unfortunately I don't know anything about Nassau County as I live (WAY!) upstate. A local gun shop would probably have the skinny on what local baloney is involved in getting a permit. Otherwise you could call County Court for your county and ask where you can ask questions about getting a pistol permit.
There is a very little amount of info on the NY State Police Website