Iím surprised your routing would bring you in from the West side if you are coming from Branson, MO. The South or East would be the best. Depending on the winter we have, May could still see snow in Yellowstone. The great advantage to May is that the bears have usually come out of winter hibernation and are actively looking to fill their bellies. You have a much greater chance of seeing grizzlies in May and June. We been to Yellowstone a few times and have yet to see a bear. Bummer!!
Fishing Bridge was the only RV park within Yellowstone that had full hookups but beginning last year and continuing into this year they are upgrading the electric so Fishing Bridge would be water and sewer only. None of the other RV parks within Yellowstone have any hookups at all.
If it were me I would plan my route coming in from the south and get reservations at Colter Bay Village in Grand Teton National Park for two days. The scenery in the Tetons is totally different than what you will see in Yellowstone. Grand Teton NP has a real alpine look to the mountains. The distance between the north entrance of Grand Teton NP and the south entrance of Yellowstone is just 5 miles. Flagg Ranch RV park is located in that stretch and would be an option if Colter Bay was filled. We stayed at Flagg Ranch last year and really didnít care for it that much, but it would do in a pinch.
It was mentioned that Grizzly RV park is not within Yellowstone and thatís correct, but it is just a couple of blocks from the west entrance. Grizzly is a really nice and clean RV park with all the amenities except for a swimming pool. Believe me, you wouldnít want to go swimming in May.
Yellowstone is a huge national park, itís larger than the states of Rhode Island and Delaware combined, so donít think you will be able to see everything in a few days. Be sure to drive through Hayden Valley and Lamar Valley for great animal viewing. Iím sure you will take in Old Faithful, while youíre there be sure to walk into the lobby of the Old Faithful Inn, it too is spectacular. Also, donít miss the Lower Falls of the Yellowstone River. Iíve put a couple of photos below this post to wet your whistle for your trip.
Grand Tetons NP, the scenery is spectacular.
An Old Faithful eruption as view from the second floor veranda at the Old Faithful Inn
The Lower Falls of the Yellowstone River.