I've stayed multiple times at the Walmart in Show Low on White Mountain Road. I went inside, made a purchase, and asked the manager if she'd mind my parking over night. I've never been turned away.
Of course, you could be turned away from the Walmart for a number of reasons. Sometimes law enforcement or management may see things differently from time to time. If you are turned away, I'd recommend you head a little farther east to the Hondah Casino RV Park. It's only $29.86 per night plus tax. It's a nice RV park with all the amenities. You can get reservations by calling 800-way-up-hi.
Lastly, you are allowed to camp in the national forest free of charge, but I'm not real familiar with places to camp right in that area. You need to be sure you're actually on national forest land that is not posted. You also need to be sure you aren't on reservation land or private property. If so, you might be asked to move by local or tribal police at 3am.
2016 Ford F350 Turbo Diesel SRW 4x4
2017 Grand Design Reflection 297RSTS
2013 Ford F350 Turbo Diesel SRW 4x4 (Destroyed by fire - 8/29/16)
2014 Grand Design Reflection 337RLS (Destroyed by fire - 8/29/16)