There appears to be no Arizona law which prohibits or restricts the possession, or particularly use of a BB or pellet gun except those generally prohibiting assaultive or dangerous conduct or criminal damage. There could be county or city laws regarding the possession or use of such devices.   

There are limits in the hunting regulations against the uses of certain types of weapons or devices for the taking of specified birds, mammals and/or reptiles.  That doesn't mean that persons under 18 may not possess or use those devices, only that such devices may not be used in general for taking certain wildlife.