I don't think horses can live on hay alone. In the wild, or a natural pasture, they browse on all kinds of things. I've laid on the ground by mine while they are grazing in our natural pasture, and watched them eat every kind of forb and grass. They even nip off the wild onions. They eat the bark off elm trees and the elm leaves when they can reach them. They eat mesquite pods. If they were giraffes, nothing would be safe.
If they are only getting store-bought hay, they definitely need a supplement. Since you are going to get a zillion opinions on what horses should be fed, I recommend you contact some good supplement providers (I use Dynamite, but there are lots of others, like Smarkpak) and talk to a consultant. Talk to more than one, because, of course, each will tout his own product. Most supplements are pricey, but with miniatures, the product goes a long way.