Horse camp is for people with horses. There is a Main Camp that is for people without horses it is dog friends and tests are allowed.
Horsemans camp is for campers with horses. Hueston woods does have another camp for non horse people. Both dog friendly as long as dog is on leash. And both accommodate tents.
You can pitch a tent, it is dog friendly as long as they are well behaved and under control. I don't know if you are required to have a horse to camp here or not. I don't see why you would want to, but that's just my opinion, take it for what you want.
Thanks! Your answer is awaiting moderation.