There are 6 (and sometimes 7) red chests in the Trash Coast. There is also a white chest which is only accessible after killing the Rakk Hive during the Another Piece Of the Puzzle mission, and which contains the third Vault Key Piece and a large quantity of cash, but no weapons. It is empty if you return there after completing the mission.
Location 1 (Red Chest)
On the small island northeast of the vendors, on the way to the Rakk Hive. It is guarded by several groups of spiderants.
Location 2 (Red Chest)
Upon entry of Trash Coast, take the path directly south of the shops. Continue until you reach a bandit camp. It's sitting near the broken claptrap. There are a number of bandits at this encampment.
Location 3 (Red Chest)
At the southernmost extremity of the map, overlooking the ocean. It is guarded by a large number of spiderants which appear in successive waves as you head south towards the chest location.
Location 4 (Red Chest)
Behind and to the south of the cage compound in which Skrappy is being kept prisoner in the Earl's Best Friend mission. There are a number of bandits at this encampment.
Location 5 (Red Chest)
Just to the northwest of the cage compound in which Skrappy is being kept prisoner in the Earl's Best Friend mission. If you follow Skrappy after you release him he exits through a small cave; the weapon crate is just to the right of the cave entrance. There are a number of bandits at this encampment.
Location 6 (Red Chest x 2 or x 1)
In the bandit encampment on the escarpment above the area where you fight King Aracobb, in the lower northwest section of the map (i.e. approximately west of the vendors). Sometimes there are two chests, sometimes only one; both chest locations are close to each other in plain view in the western section of the encampment. There are a number of bandits at this encampment.