That's a tough question. The park is very restrictive about bringing private horse in. They require Coggins and Health Certificates. They require that they trailers be swept clean, no manure or shavings on the floor. They don't like any feed in the managers. The go beyond the Certified Weed Free hay, They don't want any Non-Native plant species being carried into the park.
There are only a few campgrounds, So they focus on people in their RVs getting those campsites. The campgrounds are always full, so the park focuses on the highest demand, which is not horses.
Bottom line is that, private horses are for the most part a Day Use of the park unless you are going to do a back country camp out. You will need to camp out side the park in areas that allow dispersed primative camping. These areas are not listed as campgrounds on any list. They don't show up on any guide pamplets. There are numerous spots in the National Forest outside the park where you can do this, but they are in the lessor used areas. Away from the steady stream of RV's and Camp Trailers heading into the park everyday. Most of the public areas we use are an hour or so drive away from the park. We camp in these areas and then drive into the park each day. I have a friend with a ranch in West Yellowstone just minutes outside the west gate. We pull into his 400 acres and shut the gate behind us and turn the horses looses. But I wouldn't dare give permission or even pass on his name. It's a relationship that we highly prize being able to use his place. Because of this access, I have not spent much time looking for public camping in that area.
I can make some suggestions in the South West corner of the park and the Jackson area. Send me a PM and I'll give you suggestions and directions.
We make a couple of trips each summer into the park to ride. It's a great place to ride. Not to rocky, good footing for the horses. lots of wildlife.
We find dispersed camping in National Forest and set up a highline for the horses.
We usually just find a flat spot for the trailer, hopefull near water for the horses.
Where do you plan to ride while in the park?