It's a breathtakingly beautiful spot with a fair amount of simple pull offs for camping. Unless it's raining or just finished there won't be any water. There is precious little vegetation as well, but a fair amount of rock outcroppings to block the wind, which howls come spring and is far more common than precipitation.
Speaking of which, don't push your luck if it has rained recently, as the clay softens and becomes slick, packing around tires, and everything else. Plus even if your vehicle can handle the soft soils you'll wreck the road for everyone once the deep ruts dry out and harden.
Some sites have fire rings but that's the extent of facilities, although there might be a port-a-potty by the western entrance to the road, closer to the Moki Dugway. Summers get hot, and I'd head up to Cedar Mesa if so.
There ample dispersed camping on Cedar Mesa, as well as plenty of scraps to get a fire going. This is one of the greatest concentrations of smaller Ancestral Puebloan (formerly Anasazi, which is controversial and regarded as insulting by today's Pueblo tribes) villages, so there are a life time of ruins to explore.
Researching ahead of time will pay off, but both the BLM station by the Kane Gulch trailhead and the Natural Bridges National Monument visitor center have maps, guidebooks, and advice. Please follow the regulations, as this is a sacred ancestral site for numerous indigenous peoples.
The Recapture Lodge at the east end of Bluff has decades worth of old maps to hidden sites for those who want a hotel (I recommend them highly, having visited off and on since the 1990s). They'll give great advice if asked.
Both Natural Bridges and Hovenweap offer something precious in short supply across southern Utah, amazing and beautiful hiking, abundant ruins, and few visitors (this is the rare part). Both are far more enjoyable than Arches or Zion, which are now overrun by tourists.
Valley of the Gods, Cedar Mesa, and the nearby national monuments (also Canyonlands, which isn't as packed as the other famous national parks) offer a chance to experience Utah's red rock country without standing shoulder to shoulder with other visitors. That's getting increasingly difficult to find as the beauty of the region is well known by now.
p.s. Those wanting even more solitude should ask about the off site ruins managed by Hovenweap: you'll often have these to yourself.
