Nope, I've looked into this quite a bit. I was looking to purchase a handgun a while ago from a private party (and I'm under 21, no CHL) in Texas.
Legally, you need to not know they are a felon, are over 18, and a legal resident of Texas. I don't know what your liability is if you sell it and they have a restraining order, but better to be safe and make sure they don't.
Personally, I would get a bill of sale or some kind of receipt for your own records anyway. As heeler said, DL#, name, and address are nice. If the firearm is used in a crime and its traced to you, having 'proof' of sale is a very good thing to have.