It's a gorgeous scenic drive. Looks ideal for camping. But...
We rode the entire 17 miles on adv motorcycles, and sections of it were a challenge. There are a couple of long stretches of continuous wasboard on the west end, some rocky sections, deep dips, and tight turns on rough descending road. I would not take my car there, but a truck, jeep or suv with good suspension and clearance would do ok, even without 4wd - very little sand, not much chance of getting stuck. It's more about how much punishment the vehicle can take. Most people tend to drive in from the Mexican Hat side, exit at the base of Moki Dugway. Probably best to do it in that direction to avoid trying to share a narrow road with other vehicles.
Road was dry when we rode it, wouldn't attempt in any vehicle if wet.