Who supervises who can and cannot get a gun license in Arizona?
Nobody. If you’re not a prohibited person (criminal record; felon) you’ll pass the ‘instant background check’ NICS system. Then you get your gun. That’s nationwide, but not a ‘permit’. It’s a background check. Some states tack a waiting period on this, but that’s a state code.
Arizona, like most Western USA states, doesn’t require any say-so, approval, permit, license, or notification of the local sheriff or police force. Most US states are this way. The states where there’s a law requiring the sheriff or police to issue a ‘gun permit’ are all pretty much New England states, and Illinois.