Camping around the park is always a trick. They don't allow you to camp inside the park. None of the camp grounds are set up for horses. And of course there is no just pulling off the road inside the park. Almost all camping in that area will be Primative dispearsed camping. Meaning pull off into the trees with no hook ups.
There are several places that I camp outside the park and trailer in each day.
Turpin Meadows is just East of Moran Junction, It is the trail head for the Hawks Rest Two Oceans trails. It is a big parking area. Has 8-12 corrals that are on a 1st come, 1st serve basis. This is a good camp spot for accessing the South east Corner of the park. And you can ride right from camp.
There is a Bed & Breakfast we've used in Gardner Montana that has corrals and we have stayed there when we ride the Northern sections of the park.
The road from Flag Ranch across the South Western boundary of the park to Ashton Idaho is mostly Forest Service land. So we camp along that ride into the South Western areas of the park.
As far as rides into the Park.
In the South East, Two Oceans and Hawks Rest.
In the South West, Union Falls and Bechler Meadows
In the center - Lake Mary from the Hayden Valley ( trail goes from Hayden Valley by Asay Creek to Old Faithful)
In North East Corner, Slough Creek
Be aware that you have to carry you coggins and health papers ON THE HORSE. Any rangers in the back country can ask for them and ticket you if you can't show them.
If you drive your trailer into the park, It will have to be Swept clean. No manure on the floor. If you have any hay in the managers, You will be asked to close all windows. Yellowstone is really sensitive about you bringing in any non-native seeds.
I often camp around the Jackson Hole area. There is a good ride from Jackson Lake Lodge out over Two Oceans lake and up to Grand View.
Crossing the Bechler River
Riding Two Oceans near Jackson Lake
Lake Mary in Yellowstone
Ash Creek on the way to Union Falls
Riding in the Hayden Valley
Where are you headed in the Ashley National Forest in Utah. Being that I live here, I have rode many of those trails.