Brave the road, be rewarded with a view
If you're looking for something out of the way, but with a great view of the Tetons, this is your place.
Once you drive through the back of the Elk Refuge; possibly sighting some of the native wildlife; you will reach a wooded campground with vault toilets, picnic tables, bear proof food storage, and a site dumpster at $15 a night. There are some RV pull through sites, be sure to ask the camp host if an appropriate site is still available for your camping style.
But, if you drove up this far, have a high clearance vehicle (or good insurance on your rental car), the means to be self contained with adequate bear proof food storage, why not drive up a little further and see what else lies ahead? There are also 15 dispersed camping sites, each of them with their own special features. Some shaded, some meadow, and all of them with striking scenery. When the road forks after the campground, either direction includes designated sites, however road 30445 (left fork) is a little tamer of a drive than 30440 (right fork), but that also gets you closer to the Goodwin Lake trailhead.
On our visit in late May, the flowers were just popping. We saw a few big horn sheep on the way in and out, and a few pronghorn. There were mosquitos, but they were small and they went away once it got cold at night. The Tetons at sunset though - so breathtaking.
Take care to only use a designated camping site, store your food safely, and pack out what you brought in. These sites are all first come first serve; so have a back up plan if you come during the busy season and all the sites are full. I'm not kidding recommending a high clearance vehicle; we saw a sedan get pretty beat up traversing the ruts that had formed on the forest road. Also please drive safely - it is a steep, narrow, unpaved road. This same weekend another driver rolled her car down the canyon several times, luckily she was not seriously injured.