No; they have some limited lodging on-site, but they don't offer any type of tent camping. I would recommend looking at Dinosaur Valley State Park, which is nearby and has inexpensive tent camping and other activities, and then driving over to Fossil Rim for a tour or to do the Gosdin scenic drive in your own vehicle.
Justin is correct. Dinosaur Valley SP is only about 8 miles from FR.
Thanks! Your answer is awaiting moderation.