I think the main trick will be making the box so that all walls have to be unlocked before opening. Initially I was thinking about having the locks from each wall secure the neighboring walls... but what when only 3 walls have been unlocked, nothing would secure the wall opposite of the 4th wall.
Another trick will be making it so the box has mechanical innards but doesn't clutter the inside of the box so that nothing can fit inside. There has to be a central point of operation, some device which every lock communicates with so that every lock must be unlocked before the device will open. This I'm having trouble coming up with. It's easy enough to interlock with the neighboring walls, but it must be a common part of the box.
Latching to the top of the box seems like a natural option. But that would mean having all your locks very close to the top. I'd rather have 4 equally spaced locks on each corner of each side, and a padlock dangling down on each side. This would make a total of 20 locks, lest you add even more locks to the top of the box.
Perhaps the box lid could have inserts of its own, much like having two boxes without a top and one fits inversely into the other. This would give all sorts of options for each lock to work with, but it would mean that no lock may protrude very far into the box. The main box may have to have very thick walls to deal with this.
I think that such a project would require a lot of modifications with the locks. A friend with a welding torch may be handy.