In the Eridian Promontory, there is 1 white weapon chest and 6 red weapon chests. The first six chests are generally easy to find. The 7th chest is accessible from near the end of the Promontory, just before the entrance to the Vault.
Location 1: (Red Chest)
Just after the path turns to the left for the second time, up the slope slightly left of the main path, this chest - as well as a few ammo crates - lies under the red canopy tent.
Location 2: (White Chest)
From Location 1 (above), standing right in front of the chest and looking directly north-west, another red canopy tent can be seen. The white chest is on the ground just before this tent.
Location 3: (Red Chest)
After getting the chest #2, the main path leads due north to a broken arch near the north-east side of the map; the chest is just beyond the broken arch.
Location 4: (Red Chest)
North-west from Location #3, this chest what looks like 2 stone pillars on the next cliff over can be seen. A red chest sits on the ground in front of the pillar on the right. To get there, follow the main path, staying along the right-side edge.
Location 5: (Red Chest)
After looting chest #4, turning around and heading up the slope on the right leads to the next red chest. It is somewhat out of sight, as it is behind one of the Lance assault pods; the chest sits to the left of a Crimson Lance banner.
Location 6: (Red Chest)
Just before the entrance to The Vault, several Guardians and Lance (including a couple Badasses) spawn for the final battle of this map. There are two red-canopy tents in this area; chest #6 is under the tent on the right.
Location 7: (Red Chest)
The final chest is accessible from a pathway starting in the camp at the Vault entrance. The ledge is on the left, just north of the red tent with ammo chests. It's very easy to slip off the ledge, in which case leads to a fire pit and will force a walk all the way around. To avoid sliding requires a small jump near the beginning of the path, where the ledge narrows to almost nothing. Farther along the ledge is an invisible obstacle which must be jumped over. Once over the initial hurdle, the final red chest is sitting on a platform in the upper corner of the cliff-side.