The park's campgrounds are first-come, first-served – unless you have a group of 10 or more, in which case you can reserve a group camping spot up to 60 days in advance (for $3 per person). Campsites fill up during spring break, and several nights a week in summer, but visitors arriving by early afternoon will usually find a site.
The most convenient campgrounds are at Pine Springs, right along Hwy 62/180 near the visitor center; if it looks full, look for the 'campground host' sign for directions to overflow spots.
If all the sites are full, RVs are permitted to park overnight at the nearby state-highway picnic areas.