I live in North Central Texas outside of Fort Worth, but San Antonio is one of my favorites cities to visit in Texas.
We haven't made a trip down there since purchasing our travel trailer so I can't advise on campgrounds in the immediate vicinity, but one of my next favorite cities in Texas is Austin. I have camped there on a couple of recent occasions. I can highly recommend McKinney Falls State Park in Austin. It is just a couple of miles from the airport and basically a wilderness within the city. Beautiful crystal clear river and falls flowing through the park.
The other near-by place that I've camped recently is Jim Hogg Corp of Engineers park at Lake Georgetown. Every (or most) site has a large covered picnic table area. All the campsites are spacious.
I'm sure that others can advise of campgrounds closer to San Antonio, but Austin is also a great place to visit.