I'm coming up from Utah. So we either drive to Jackson Hole or West Yellowstone. I have friends with property in both areas. Some trips when I come with them we just pull through the gates of the ranches and turn the horses loose and collect them in the morning. Other times when they have family or larger groups up, we primative camp and just meet them at breakfast before heading into the park.
The park itself doesn't allow any dispearsed camping. But the forest service land around Yellowstone does. So you will have to trailer outside the park and pull into some areas managed by the Forest Service if you want to camp on public land.
There are several trails that I have ridden and enjoy. Several have trail heads just outside the park where you can park and then ride the trails into the park, others you trailer into the park and then ride.
Slough Creek in the North East section of the park.This is a 7-8 mile long trail that follows along the edge of Slough Creek. It passes through some beautiful meadows. It is mostly used by fly fisherman who hike up the trail to fish the upper reaches of Slough Creek. It's a fairly level trail with good footing. You access it from the Slough Creek Parking lot which is just off the road coming in from the Cody park entrance.
Mary Lake trail is in the Hayden Valley. We call this our Buffalo ride, Because you will always see lots of buffalo in this area. It follows up Asay creek to Mary Lake. I guess the trail goes all the way over to Old Faithful, but we have only ridden the 10 miles into Mary Lake. In and out makes a 20 mile round trip. You cross a lot of springs that spring out of the hillside and run into Asay creek. It can have a few boggy areas if you ride too early in the season. It's usually an August or September ride for us. This trail starts right on the main hiway between Canyon and Yellowstone lake, You park in one of the pull offs on the side of the hiway. It's kinda fun, as we saddle up and get ready to leave, many folks see you from their cars as they drive by. We have many many folks pull over and come ask to take our pictures. This area is mosty rolling hills. Again it has excellent footing and great place to trot or canter along.
South East Corner of the park has numerous trails. These are accessible from the gravel road that runs along the southern edge of the park from Ashton Idaho to Flag Ranch just outside the South Entrance to Yellowstone. Some of the more popular trails lead into Bechler Meadows or Union Falls. Most of these trails need to cross one or more rivers. We usually wait until the snow melt has subsided and the rivers have dropped to lower levels before we ride these trails. Somebody always seems to come off their horse while crossing the rivers.
These trails are much more wooded. Not the big open country like the northern side of the park
But they are well worn and easy to follow
South east corner of the park.
Turpin meadows is a popular trail head. It has room to park 30+ truck/trailers. It has some corrals, water hydrants for horse water etc. From the Moran hiway take the buffalo river 7 mile long gravel road. Turpin Meadows is the end of the road. This trailhead is used mostly by packers going into Hawks Rest or riding the Throughfare trail.
South of Yellowstone we also ride alot in Grand Teton Nationa Park. From Jackson lake Lodge you can access Two Oceans trail and ride up to Grand View. for a spectacular view of the Grand Tetons
This trail rides along the edge of Emma Matilda lake and around Two Oceans lake.
I know there are some trails that come into the North West Corner of the park, off the Gallatin Highway between West Yellowstone and Bozeman, But we have never ventured that far north. But I'm sure with a little reseach you could find info on those.
What I have noticed is that folks from back east are looking for horse camps with hook ups etc. These are just not very prevalant here in the west. Most of the locals just primative camp in dispearsed camp sites. This is unfamilar to folks traveling from the east where there are no large tracts of public land. They want to call ahead and make a reservation for a space and are frustrated when they can't info on where to stay.