In Free Mode you can buy buckets from Black (the tall female with black hair and red feathers), she is located in the same area as the player's home (go along the side wall and you'll bump into her eventually).
To access her shop: interact (LMB) > Shop (select from the popups) > Change to Items shop tab (tap E) > go down and select the bucket to purchase (WS up down, enter/LMB to purchase).
QuickTip: Empty buckets can be carried faster, after interacting with the bucket, tap 1 (you can also see the quick slot command at 1). To get it out, just throw the bucket as any other item (quickSlot 1/2 > RMB). To equip the bucket it must be in carry state (can't do this while it's stored in slots).
SideNote: Money can also be made via SlutJob or by taming-breeding-selling slaves. For more info please check PauseMenu [Esc] > Options > Help > go left right thru help screens until you find the desired one.
I hope this helps,
~NK