Depending on the type and quality of the mailbox, it's probably made such that any of the following are possible:

• getting a new core for the lock
• drilling or modifying the lock so it no longer needs a key
• making a simple key (if the lock is fairly simple, so is this option)

If in the unfortunate event that the lock is decent and the and the mailbox fails without the lock in place and the lock's not replaceable, you may need a new mailbox.
All of this assumes that you own the mailbox in the first place. If not, then all of this is your apartment managers responsibility and (s)he'll take care of it for you (and possibly bill you for what should be reasonable costs).