You’ll often encounter chests and doors in Fire Emblem Three Houses that are locked. Getting into them requires one of two things. You either need to find a key on the map dropped by enemy, have a key that you purchased, or have a unit with a lock picking ability.
You’ll want to loot every chest you find in Fire Emblem Three Houses if just for the potential for powerful weapons or armor that could be inside. If there is a chest on the map, there has always been an enemy that holds a key to it in our time with the game. The same applies to doors.
The problem that you’ll run into is that sometimes certain levels have move requirements, or the enemies that are guarding the chests could be more powerful so you have to make a call on whether to just avoid it altogether.
However, you can avoid looking for a key by purchasing one from the marketplace. The item shop will sell both Door and Chest keys to keep on hand in case you run into a situation where you can’t find it on the map. They cost 300 Gold a piece.
Alternatively, you can create a character in the Thief Class. The Thief has unique abilities that allow them to pick locks with the Lock Touch ability. If you are looking to create a Thief Class you will need to pass a Certification Test once you’re character is a high enough level and has the correct stats in place.