I'm not aware of any bank in Avalon.
-You could talk to Barlow, that's the man who sells everything and stores the crystals you find.
The shopkeep is the closest thing to a bank where he holds onto your crystals you collect. He's by the castle keep.
Except for the fact that today, when I went to Barlow to add 300+ crystals, the previous 1500+ have disappeared! Someone embezzling?
Looks like you'll need to buy the dungeon expansion from Barlow to place him in there till you get your crystals back.