Yes, you can camp in the Painted Desert, but options are limited since camping is not allowed within the boundaries of the Petrified Forest National Park, which includes much of the Painted Desert. However, there are a few options for camping nearby. The Painted Desert Community Complex, located within the park, offers a few RV sites with no hookups, but these are only available for self-contained RVs.
Outside the park, you can find some private camping options on Hipcamp, such as the Painted Desert Ranger Cabin and Painted Feather Farm. Additionally, you can explore nearby campgrounds in the surrounding areas, like Holbrook, which is about 25 miles from the park.