Yellowstone opening & Closing DatesLooks like the Old Faithful Snow Lodge and the Geyser Grill are also open now. The Geyser Grill is cafeteria style eating. We have used the Snow lodge a few times without reservations, but usually had to wait for a while. Those times were in June though.
The Mammoth Hot Springs Hotel is also open. Been there a few times and never had to wait for a table. Food was good.
Looks like the Lake Hotel (big yellow building) isn't open yet.
You are getting there well before the serious tourist season, so I doubt there will be any need for a reservation at the Old Faithful Inn dining room.