There has been an effort underway for quite a long time to gain approval for a regional firing range on National Forest land in the near vicinity. It isn’t going to happen any time soon.
I suggest you become accustomed to picking a safe spot in the National Forest and enjoy yourself. Just be certain to haul out empty boxes, brass, targets etc. Leave it cleaner than you found it.
Some Flagstaff residents I know drive to Holbrook and/or Payson once a month to participate in IPSC shoots. There are other places in Arizona but those are the closest organized shoots I am aware of to Flagstaff. Holbrook is 70 Miles and Payson 100 miles from Flagstaff.
Payson has a great range, accomodating pistol and rifle out to 300 yards, about 6 miles out of town to the south. It's run by the Tonto Rim Sports Club of which I am a member. Payson has real IPSC shoots, Holbrook has IPSC style shoots. Both are fun.