So the Fatman weapon dropped from an awarded lunchbox and I was wondering if using it inside the vault could cause any friendly fire or anything like that.
Nope, the Fatman is safe to fire inside your vault. Just like other area weapons (missile launchers, flamethrowers), it does not damage other dwellers or the rooms. In fact, none of the weapons damage other dwellers or the room itself. Vault-Tec sure does make those things sturdy!
Fatmen are a bit slow to aim and fire; you won't get more than a couple of shots off at deathclaws, but it will take raiders, molerats, and radroaches down very efficiently.
Fatmen look like they are slow to aim and fire, but enemies will still take damage the entire time the running aorund the room/aiming/firing animations are playing. Source: Two Deathclaws dropping dead while six fatmen toting dwellers ran around the room trying to get their shot lined up. No one actually got around to firing the fatman before the surviving deathclaw fled the room.
When a dweller finally does get a shot off, it looks pretty cool.