Do horses naturally jump?

Yes and no. Horses are born with the ability to jump - in the wild, if they're running, they need to be able to get past/over obstacles. While they'd prefer to go around, and will generally step rather than jump if they can, they certainly can and will jump in the wild.

However, they do have to be taught how to jump carrying a rider. And not all horses "naturally" enjoy jumping - in fact, a horse that really enjoys jumping is probably the end result of selectively breeding horses that enjoy it. (Some horses seem to think it's a great game, and others would rather do something else).