|Subzone||Village of Outcasts|
The Purple Chest becomes available after you rescue the Blacksmith. It can be found in the Dark World equivalent of their house. If you have the Magic Mirror, you can walk to the west of the entrance to Swamp Palace and Mirror back to the Light World to give it to the Middle-Aged Man to open.
Note that pressing the A button (even to dash if you have the Pegasus Boots), will drop the Purple Chest on the ground and you'll need to retrieve it from that point to continue. If you lose it or can no longer access it, if you Save & Quit and it's not following you, it'll return to its original spawn point.