Never rode the FLAG area but I have been there lots of times. Flag has an interesting dynamic. Lots of trees and rocks. Most of the land is state parks or forest/Tribal land that you can't ride on. To the North East there is a state park which is on a Volcano eruption site, you can potentially ride the South Eastern side of the Volcano just outside the park seen lots of dudes riding dirt bikes and sand rails up there. I really doubt however, at least all the times i've been there (10 plus times 4 years in a row) That you would want to consider off trail riding. There is also a ski resort 20 min west of Flag that goes up from 7700 ft to well over 12,000 but you cant ride that either.
Most of the People that Ride in AZ ride a little group of mountains East of Phoenix called the White Mountains.
Then you have the Jacob Lake trail system, no off trail riding but they are supposed to have some OK trials.
AZ definitely gets enough snow to ride, but its mostly trail riding unless your going to drive to the white mountains which is 6 + hours away, at that point for my money if I were living in Flag id drive to Southern CO our UT. Lots of dudes from New mexico and further south that you could meet up with and ride. Post in the state sections im sure you will find some people to go with.
I know your post relates specifically to snowmobiling but Arizona and Flag is great, the bar scene, hiking, fishing, tourist stuff, river rafting, ect is unreal. Id move to AZ yesterday if the opportunity presented itself.
