The Jagdhof is right off the main road so if it is passable you will not need chains or studs. That road goes right through the town and carries all of the local traffic heading towards Salzburg. The Schloss Fuschl is down the hill and might have different requirements.

I'm curious how the new buildings are, they were just under construction when we were there. I really enjoyed the hotel when it only had the one building and 24 rooms.