Yes, camping is allowed at White Sands National Park. There are 10 backcountry camping sites available, which are accessible via a short hike from the trailhead. These sites are on a first-come, first-served basis and require a permit that can be obtained at the park's visitor center. Keep in mind that there are no campfires, pets, or vehicles allowed at the backcountry sites.
Additionally, you can find uncrowded camping options just outside the park at Hipcamps.