I've made payments on a couple horses over the years. Usually from an acquaintance or friend of a friend type deal. I wanted the horses. They were for sale. I worked out payments. 2 came home with me while I paid them off. 1 stayed with the previous owner. I could afford to keep, vet, feed, and care for them. I just didn't have that much cash on hand.
I won't touch the vet fund for anything but the vet. The household emergency fund, same thing. People make payments for a lot of reasons. It doesn't necessarily mean they can not afford to own a horse!
OP I really hope this turns out well for you. I have my fingers crossed for you!