While the parks may have fire rings or charcoal grills fire restrictions may make them moot. When we were out there last spring if it didn't have an "off" switch it could not be lit outside. Gas grill fine, charcoal not fine. Private park or national park or national forest same rules.
We stayed at the Flagstaff KOA on the north side of the city a little spendy but a great location and a nice well shaded park, decent showers. Had a fenced in off leash dog area too if that is of interest. A food wagon that sold
hot breakfasts was also available if one wanted to grub and get going with less hassle and clean up.
Sunset Crater Volcano national monument off of 89 and the Wupataki National Monument made a very nice day trip and were well worth it. I think there was camping both dispersed and in the park near the visitor center park office where we bought a day pass. I didn't camp there or really inquire on what was available so you would have to do your own checking to confirm availability, costs, or amenities. I know there were some sites going into the Crater national monument that were just cleared spots off the road typical of national forest camping.
Short drive up 89 to 64 west to get to the south rim from there if one didn't want to stay at the KOA and didn't managed to get reservations at the canyon proper.
I liked the north rim better but the south seemed to have a great deal more overlooks and pull overs by far. The elevation of the north rim made for cooler summer temperatures and it had forests when I was there many years ago. Went horseback riding through them to a couple of overlooks. South rim is pretty arid with a lot of people and many places worth stopping at. One can spend a day just working ones way along the drive through the park.
Yes push come to shove I can always Walmart camp or whatever but if I'm going to drive 2000 miles each way to get to a destination I just might want to make a reservation in order to insure being in the location I want to be at. If I end up having to wing it I'll survive but for that much gas money and effort I like to tilt things in my favor.
I often do not make reservations for the travel days until later in the day of. If we are feeling like driving longer or tired early that determines when we start going online and locating a campground. Get some real winners and loser that way but none have been so bad I didn't survive it. Worst case I sleep in a truck stop and get a hot breakfast or coffee on my way back to the road in the morning.
Traveling with brother in law I'll do a rest area for the night, traveling with wife I tend to avoid them. He is ex airborne ranger, good solid backup to any trouble. Wife has some street smarts but doesn't provide his level of "second thoughts" to people up to no good. To petite and not at all intimidating, unless you forget to put the dirty clothes in the hamper.... where they DO belong after all.
BTW - be advised the moose that seem like tall cows are not. They can have a temper and are fast.