I doubt Beth or Morty will be doing any horse fucking. That said Beth is a horse doctor so I imagine that will play a roll in the event.
If I had to guess, some VIP horse person is mortally wounded or sick, Beth has a crisis of confidence or some other reason why she can't/won't initially help. Morty steps in and helps Beth get the confidence or in some other way facilitates Beth being able to save the horse persons life and be a hero to the horse people. Riding the high of their adventure the two fuck like rabbits with Beth no longer caring about making Morty ware a condom.