There are a number of options. These are all FS campground since I won't recommend the hustle bustle of the NPS campground and they could be hard to get into unless you get reservations in advance. Although I work with a number of great folks from NPS we still affectionately call them the Park nazis :-laf
If you are not pulling a TT you can stop at the Sheffield FS campground just past the Flagg ranch and as you cross the Snake River heading south on the east side of the hwy. This campground was burnt over in 88 so there isn't much shade if it is hot, but it is only $5 and you probably won't see anyone else except for the outfitter about a 1/3 mile away. The Clovers offer day rides and are an old time family and can definately fill your ear with all the earliest, latest, and future info

-- good folks.
Next there is Pacific Creek. After you drive up the GTNP road to the forest boundary you can use a "dispersed" campsite of your liking. No fee, but there is a stay limit of 16 days. At the end of the road, which is a portal to "my" country, there is a campground under construction with toilets, water, public corrals, and such. Right now it is officially a trailhead but there are plenty of camping areas. Won't be too much traffic here except for folks driving up and a few day riders, BJ Hill offers day rides and wilderness trips about 1/2 mile before the trailhead. Great close knit family run operation.
Then along the Buffalo River Road there is dispersed camping at several pullouts. Then there is Box Creek campground ($10) that is about a mile before Turpin Meadows. There are some switchbacks, so not recommended for TTs although you can pull a horse trailer up the hill. Not much traffic here. Next is Turpin Meadows campground ($10) that has corrals, toilets, water, and a lot of traffic (nay, mongols hordes reminiscent of "J"enghis Kahn as John Kerry would entone) heading into to be disappointed at the fishing in Yellowstone Meadows. There are a number of outfitters along the Buffalo road who offer activities such as day rides, fishing, wagon rides, and cookouts including the Castagnos, Heart Six, Wagons West, and Turpin Meadow Ranch.
About eight miles east of Moran on Hwy 26 is the Hatchet campground ($10) or you can drive up the Flagstaff road to a dispersed site.
Lastly, there is my place (free) which is 8. 5 miles east of Moran on Hwy 26. It's not really "my" place since it is a FS house, but who else could afford to live here at $200K an acre??? If I'm not working back in the wilderness or out on an adventure, I'd be happy to let you park at my place. No TV or phone, two very friendly border collies, and I live like a bachelor since my wife is in BZ. But I do have showers, spare rooms, and a gift of gab.
EDIT: Just found out the Turpin and Pacific campgrounds will be intermittantly closed for reconstruction. However, what is available will most likely be free instead of the $10.