There are only NPS campgrounds without services and only a few of those. There is a State Park north of Padre Island at Corpus Christi on Mustang island.
For RV Parks you have to go to Port Arkansas. The nicest of those is Gulf Waters RV Resort. They are an owners park that rents to visitors. They do have rules about what is allowed in.
What are you looking for? What kind of camping? Have you been to the area before? It’s pretty hard scrabble, certainly not like Florida beaches, buts it’s a great area to visit if you’re not expecting “resort” amenities.
Keep in mind that Spring Break is a thing there between mid-March and mid-April and it can be full up.
2017 Winnebago Adventurer 37F
2016 Lincoln MKX Toad