Horse & ponies are sometimes available as a “Care Lease”. There is no payment to lease the animal but you are required to pay for all the expenses for care of the animal. This is done sometimes with a young pony where the owner wants to keep the pony but wants miles put on him. In those cases owners may require the pony be shown x amount of times. Some owners don’t want to sell but cannot afford the expenses.
Half Leases can also be done and may fit your needs. Basically two people lease the same animal or the owner retains the right to still use the animal half of the time. This is a great way to save money and works best if the riders are at two different skill levels or ride different disciplines. You would only have to pay half the expenses. The only downfall is when both riders want to use the animal at the same time.