I think much of the water they ingest comes from the food they eat. Almost everything they eat, or prefer to eat, is a live plant. Most plants need to be very hydrated to survive.
On hot days, deer need more water than these plants can provide because they use more than normal. Perhaps part of the reason we see so much sign near water sources is because of the thick, lush vegetation that surrounds these water sources. I know the ponds and creeks that are on the land I hunt are choked with apple trees and honeysuckle. Both of these plants/fruits contain a large amount of water.
These are just my thoughts based on the fact that I have never actually seen a deer drink either. I know that if I ate apples and berries all day long, I wouldn't really need to drink very much water either.