Oh no!! The word is out on Bonito. :(
Seriously, this is a national forest campground and it doesn't take reservations. It serves mainly as a transient campground for those passing through and just want a night (although we always stay longer). If you can arrive by noon you'll most likely get a site.
Flagstaff has a few regular RV parks but they aren't great. I would think if you call a few days prior you'd get a site.
However, I wouldn't try to stay in Flagstaff for touring the Grand Canyon or Utah parks. It's much too far. Trailer Village IN Grand Canyon NP is the place to stay and then move into Utah. Have a good trip!