My guess is that the horse helps her stay sane more than you know. However maybe a break from riding/competing may be a good idea until she can overcome this. But NOT separation from the horse.
This type of temper is a lot more common than one might think, and not nearly as black and white as people assume. Therapy would definitely be the #1 choice. I'll bet she and her parents will be open to it, assuming they all aknowledge the problem and it is kept between you and the family (publicizing it would probably be detrimental).
The girl needs to develop a new coping strategy for her anger, and the parents should be there to help (gently).
As someone else suggested, maybe even an abusive home life or boyfriend. One of the most common factors in youngsters who throw tantrums like this is an abusive home life. Just because the parents appear nice doesn't always mean it's the same at home. I know it isn't your job to raise this child, but simply having someone that she can open up to may fix a lot of problems.
Good luck OP.