Can horses get epilepsy?

Yup. It's not that common but, of course, a thousand pound animal having a seizure is not something you want to be anywhere near it. Horses can also get seizures because of toxins, trauma, cancer, or inflammation in the brain.

There are drugs that can be used to control seizures. Needless to say, horses that are having seizures regularly should not be worked until the condition is under control.

A form of inherited epilepsy is found in Arabians - this condition starts at about 6 months and goes away at "about two years" - probably at puberty. Because it goes away before the horse starts being worked, it's only an issue if they injure themselves.