Haven't camped there, but about seven miles past Galveston SSW, there is a very nice campground called Jamacia BeachRV Resort. All there 100+ sites are pull-throughs. Nice owners. Right across the two lane highway is the beach. I've driven on it while fishing.
I know camping is allowed on the beach. We've seen large fifth wheels on North Padre Island where the beach is hard as a rock. The Galveston beach is the same. Never been to the peninsula but I can't imagine it would be any different. There are a couple of campgrounds in Crystal Beach if you get there and see that you'd rather not give it a go. You do need a beach pass however if you're going to stay on the sand. Good luck.
2012 F350 KR CC DRW w/ some stuff
2013 36FL Cindy and Tom, Toby and Kasey (our Berner and Newfie)
Oh...I forgot the five kids.
We are actually staying at the Jamaica Beach Rv Resost on Galveston Island for a month. It is a great park with very friendly people from all over the US and Canada. If you camp on the beach in this area you need to be very cautious of tide levels, because the beach elevation is minimal.
After seeing the amount of damage done to the Fish and Wildlife trailers parked down on the Texas coast, I would never take our Redwood on the beach. Even their trucks that make runs down to Boca China TX get destroyed with salt and sand. This includes all suspension components, brakes and wheel bearings.