What happens if horses don't have grass or hay during winter?

This depends on what else the horse may be getting to eat. If the horse is on a complete feed it will probably be OK, but it is always best to provide hay or grass whenever you can as this is what horses evolved to eat. If hay is scarce or just not available look into possibly feeding oat straw in small amounts daily. You could also try to find hay cubes or hay pellets. It should be noted that hay or grass is what enables a horse to maintain it's body heat in the winter through digestion of the long stemmed fiber in the hay and grass. Without these things the horse may need to be blanketed or even brought inside, particularly if it does not grow a very thick winter coat.