Most, if not all caving around Carlsbad is within the Nat'l. Park Service boundaries. Permits are required from the cave specialist office at the park. Ridge walking also requires a a permit.
The head honcho is Dale Pate, a good guy from Texas. Also in the office is Tom Bemis and Stan Allison, both longtime and somewhat legendary cavers of the region.
Camping within the park? None, but it is available down the road in White's City or the town of Carlsbad. But west of the park is Lincoln Nat'l Forest with campgrounds. It is a long drive, though.
Thirty five or forty miles down the road toward El Paso is Guadalupe Nat'l. Park. Numerous caves there as well, but a permit is necessary there too.
Hope this helps.
Have fun.