There are 208 first-come, first-served sites at Twin Peaks Campground by the visitor center. You can probably find a site here, even in high season. The campground has drinking water and toilets, but no showers or hookups. Tenters might prefer the scenic, if primitive, Alamo Canyon Campground, which requires reservations at the visitor center. There's no backcountry camping because of illegal border crossings.
Try removing some filters. Alternatively, there are some nearby lodgings with availability below.