New York does not require a permit to purchase for long guns. There is a permit requirement for handguns. Permits for the purchase of handguns are given on a "shall issue basis". Thee is a fee for every handgun that is added to the permit.
See New York's state description for information on New York City
Some states require that a citizen acquires a permit, before they purchase a gun. If it is required, there will be a fee that varies from state to state. Some states also require finger printing, and background checks before issuing the permit.