What part of NY? Each county has some slightly different rules.
In general, you need to have a license to posses a handgun in New York - even in your home. The good news is that most (almost all) of the licenses issued are to carry concealed, although the issuing judge may add his or her own restrictions to the license. NYS law makes no provision for restrictions, and as it was explained to me by a State Trooper, the "restrictions" on the license are the purposes for which the license was issued, not legal limitations on how or ware the handgun can be carried. Although NYC and Buffalo are off limits.
I believe the right way to do it would be to have the guns shipped by a FFL in AZ to a FFL in NY who will hold them until the license is issued. Alternatively, he could probably leave them in someone's care in AZ and then transfer them into NY after the license is issued. A word of caution though, all the handguns in your possession have to be listed on the license. NY likes to know what kind of toys you have.
It amazes me that with such a simple and easy to navigate system involving about $200 in costs, a thorough background check, lots of paper work and running around, a 3 to 6 month wait and multiple trips to a notary, criminals still would rather buy a gun on the black-market instead of just following the rules.
Here is a helpful website that explains the process in my county:
Orange County Shooters