I doubt they do. I don't think stables are a common thing for state universities to have. You can check their website though and that will tell you for sure. I wouldn't worry about it right now. You are young and have years before you will need to worry about college. You could always live at home or lease your horse for the time you are away.
Even if there are not stables directly related to the university, there are bound to be plenty of boarding options within the community surrounding it. I would not discount your education entirely based solely on being able to take your horse with you, jmo.
I'm completely unfamiliar with MSU and also have not checked out their site but I'm also wondering if they have local boarding facilities who advertise for boarders somehow through the school. Kind of like off campus housing for horses of the students in the equine programs. Sounds like it would be fairly simple to ring them up and ask.