Do gun owners in Massachusetts have to go to the local police station every four years for mug shots and a full set of fingerprints? Do they also have to immediately notify the state authorities whenever they move?
To find out rules in each state, go to the attorney general websites of that state. The laws change frequently.
The NRA has summaries of each state’s laws. As they said quite up front on the page, this is a summary only and the law changes all the time. Don’t trust this any farther than you can throw it.
That said, take a look: