Generally speaking, the Lexington KY is a HUGE area for horse breeding. Obviously primarily Thoroughbreds (which are only live cover, quite different from AI breeds), but if you want experience with breeding, mare management and foaling/foal care the TB farms have it in spades and will be some of the more intensively managed farms out there. The is a wealth of knowledge and expertise, and such a high number of foals (and work!) that many have breeding season internship type positions. Obviously no riding on race horse farms though. KEMI is a internship program that helps people get positions on TB farms in their area of interest: http://www.kemi.org/
There’s also a growing amount of WB breeding in the area, since Lexington has such a wealth of resources. Spy Coast Farm comes to mind as a h/j breeder. They’re a bit unique in that they also have a young horse training program and sell many (most?) after they are under saddle. No idea who rides them though or if they have working students/interns.
Other option would be to go to Europe, since you’d need some sort of visa whether US or EU, tons of jumper breeding there (obviously no hunters, that’s a N American thing) and generations of expertise and knowledge.