The Bronze key is an item that players can steal from H.A.M. Guards in the storerooms (lower level) of the H.A.M. Hideout.
After using the key to unlock the chest, the player receives a message in the chatbox saying: "Your bronze key breaks in the lock." and the key disappears from the inventory.
All of the H.A.M. keys (bronze, iron, steel and silver) can be obtained at the lowest level of the H.A.M. Dungeon by pickpocketing the H.A.M. guards. Rewards from the chests include: coins (small amount), 1-5 pieces of random jewellery (ranging from sapphire to diamond,) and/or cut gems (also ranging from sapphire to diamond.)
For an exhaustive list of all known sources for this item, see here