When you go on missions in Avenger's Initiative, pressing up on the D-pad reveals a bunch of questions marks located around the map other than just the main objective. If you just go towards the main objective you will miss a lot of these so take these solo or a team that will go exploring with you.
When you go towards these, Jarvis will announce what the question mark is. The one you are looking for sounds something like "There is a chest with gear in your area."
These chests are the white boxes that are seen in the achievement image. The resource chests (golden lid) and the regular chests (bronze lid) do not count for this.
Some of them are located in the open and some require you to open depots or other doors. These open in on of three ways and can be grouped together in the case of depots.
A) Shattering a glass door with a heavy attack with Hulk, Ms. Marvel, Thor, or Iron Man in the Hulkbuster.
B) Keypad that only Black Widow and Iron Man can open.
C)A depot with a timed button pressing puzzle. Some are yellow pressure pads that you have to stand on. Some are white consoles with a yellow hexagon against walls that you have to melee until they turn blue. Some are yellow gears on walls that you have to shoot.
When you hit all required switches (usually 4 but a few in Eastern Seaboard are just 1), they will turn green and the door will open.
I would suggest Vault missions (blue icon in war table) as they offer a lot of chests and are a good way to get this quickly.