They don't have any problem with it. In fact, that's what most hunters do. Here's some info straight from the Forest Circus website:

Dispersed Camping(other than in a developed campground) is allowed at no fee and without a permit in most areas of the Forest. You are allowed to drive within 100 yards of an open road to set up camp provided it will not result in damage to the land, and the area is not signed otherwise. Dispersed horse camping is allowed throughout the Forest. If camping is not allowed, signing will be posted.
The following are some dispersed camping tips:
[ul][*]Please camp at least 100 feet from lakes and streams.[*]Please bury human waste at least 6" deep and 200' from water.[*]Leaving any type of cache on National Forest is prohibited.[/ul] Please pay special attention to the following regulations. They allow everyone an equal chance to enjoy the National Forest: From June 1 to September 10 certain restrictions apply:
[ul][*]Camping is prohibited within 1/4 mile on both sides of designated portions of Forest Road (FR) 26 (Big Goose), FR 125 (Little Horn), FR 16 (Black Mountain) and FR 137 (Porcupine Creek). Watch for signing.[*]Camping is prohibited within 1/2 mile on each side of Highway 14 and 14A.[*]Length of stay is limited to 14 days at any given campsite or immediate vicinity, which includes the leaving of any type of equipment, tents or trailers, from June 1 to September 10. After 14 days, you must move a minimum of 5 air miles. You may not return to the original site for 14 days.[*]Storage of camping equipment for longer than 14 days is allowed only at designated storage areas.[/ul]The campgrounds do close in the fall, but most of the time you can still set up camp there if you want. The water pumps and outhouses will be locked, though, so bring plenty of water. None of the campgrounds in the Bighorns have electricity hook-ups. You'll be glad your camper has a generator- especially if the heater runs off a thermostat. It would be helpful if you could leave the camper at the campground when you take the truck to wherever you plan to hunt, though. Some of those mountain roads can be a little rough and you never know when you're going to need to drive under low hanging tree branches. Just a thought.

If you do decide to fly out, there are several lodges on the mountain that you might look at. The most popular three are located around Burgess Junction (just outside your hunting area), and there is a lodge on Meadowlark Lake between Buffalo and Tensleep on highway 16. There are also hotels down off the mountain in Tensleep, Worland, Greybull, and Lovell.