During the course of Chapter 7, you will be required to activate consoles using keycards relating to the Airbuster weapon.
There are 3 rooms containing consoles and after the third room, you will continue on until you go up the elevator and in the final room, you will be prompted to start moving levers along with Barret and Tifa. Instead of immediately helping them, turn to the console to their right and activate it.
Then join them in moving the levers and the door should open, unlocking the trophy and some loot.