FFR776 is the main access road from US-89 to the Cinder Hills OHV area and FR777. Camping is prohibited until you get into the Hills. There is no through-access to the Sunset Crater National Monument because a closed gate at the end of FR776 prevents you from connecting. There is excellent RV-accessible camping and group camping along FR776 past the sign into the Hills.
People come here for a one-of-a-kind experience: climbing "dunes" made of volcanic cinder at 7000-8000 feet elevation, with amazing views. This is a really unique and excellent attraction. It is also not for the faint of heart or those with low clearance or street tires. Most users come to camp along FR776 and tear it up on quads, razors, dirt bikes, and rails along FR777 and offroad. Some of us, more casual offroad enthusiasts, show up in jeeps or something similar - and will not be climbing these epic slopes. Those who come in RVs will stay close to the main road FR776 and around Cinder Lake. This OHV area is right in Flagstaff, so you can get out, have fun, and return to your home or hotel in time for lunch. This is also one of the best places to camp in Arizona, especially for large groups, because of the unlimited and uncrowded dispersed camping options.
Sunset Crater National Monument has two campgrounds (Bonito and O'Leary Group Campground), but these are on the Sunset Crater side of the area, not the Cinder Hills OHV side. Dispersed camping is allowed throughout the OHV area and is abundant. There are many excellent large-group RV sites along FR776. Camping in the Cinder Hills is fantastic! There are no bugs, and it's such a large area that it never gets crowded.
The astronauts of the Apollo moon landing missions trained to drive their moon rovers and use their equipment here! Cinder Lake, which you see on the east side on the way in on FR776 just inside the Cinder Hills OHV fence, is where that happened. NASA detonated explosives all over this "lake" to form a series of craters, and the astronauts practiced their moon rover-4x4 driving. You can still see some of the craters using satellite photos overhead. When you head across that Cinder Lake in your jeep, you're re-enacting a significant part of U.S. history! According to NASA, this is as close as it gets to driving on the moon. When you're out in the middle of Cinder Lake, it's not hard to imagine driving on the moon or Neil Armstrong training here in the 60's. Head up to the USGS Astrogeology Center in Flagstaff for more of this interesting history (you might need an appointment). Or, visit Lowell Observatory in town, where those same astronauts worked with scientists to map the moon's landing zones using the Clark Telescope there.