Several of the campgrounds that I am aware of around here are closed, but it is only if they are in the Stage 3 restriction areas. Pine Grove and Ashurst Lake campgrounds, off of Lake Mary Rd are open, Bonito CG, near Sunset Crater is open, but Canyon Vista is closed. Stuff immediately south, and north of town is all closed. But, there is still dispersed camping as of Saturday am, in parts of the Coconino and Kaibab are surprisingly still under only Phase 2 restrictions.
A late reply for the OP (hope you had a great 4th), but still relevant for folks visiting the high country...
What Pitch and EricB1 said. Dispersed camping is your best bet - lots of great options - but only outside of closure areas. For closure areas and fire restrictions otherwise, see: Coconino National Forest - Home
Instead of closing the entire forest, the USFS this year has only closed high use areas where fires would be especially destructive. If it works, that's a good compromise for everyone. Let's prove that strategy successful by following fire restrictions in areas still open.