Guadalupe Mountains Campgrounds
The park 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).
If you're planning on sticking around for a night or two, there aren't a lot of options. You can drive 45 minutes to Whites City, NM, or you can camp in the park and bring your own food. Aaaaand…that's it.