Hello Lily !
I think that you need to have quite good results in shows, or to have a special project with your horse, such as an original Youtube Channel, a nice Facebook page... The point is, you need to be interesting for them in the sense that you would allow them to gain visibility. Then, you can either be contacted by them or either try to reach out for them, but in this case, you'd better have a nice file with arguments on why it would be interesting for them to sponsor you.
Good luck for your project ! Have a nice day :-)