There is a primitive campground within the park boundaries now. It's located on the southern side of Rain God Mesa. Have not camped there myself, but have noted the sign for the campground during multiple trips there this past year. I checked the Navajo Nation website and it has no info on this campground. You may want to contact the Monument Valley Visitor Center directly for the most accurate info ...
Also, the link to a map is broken. In labeling the photos I uploaded yesterday I did a lot of searching for maps that identified the geological features. This was the most complete one I could find--it's partly in French. If you wish, I could fix that and upload a copy.