Yes, you can car camp in Great Sand Dunes National Park and Preserve. The park has a campground called Piñon Flats Campground, which accommodates tents and RVs. There are no hookups for RVs, but the campground has restrooms, potable water, and picnic tables. You can find more information about car camping in Great Sand Dunes on Hipcamp.
Additionally, there are private campgrounds and Hipcamp sites near the park that allow car camping.