Warhammer 40,000: Inquisitor - Martyr

From my experience, the red Medical Chests do NOT count toward this achievement. Any type of loot chest does not count, as well. Only the green supply chests that refill your belt item that you find during missions count for this achievement.

Open them even if you are full and cannot pick up the contents. It will still count. All you need to do is open them.