Ugh, do they. Horses can and do get ear mites, although they are rare. The most common culprit is Psoroptes cuniculi, a mite more commonly found in rabbits (and presumably transmitted by wild rabbits that wander into the pasture).
If a horse does get them, though, it can be a real pain. Because their ears hurt, they don't want to let you near their ears to kill the mites - they often have to be sedated. I know at least one horse who had a major infestation and still doesn't want to let anyone near her ears. And I heard of another that got so ear shy as a result they have to be bridled like a mule!