Since this is a rather unique situation, I just thought I'd share my experience in obtaining a drop box key in case others have to go through this experience.
* I needed a photo ID and the Church's bank account number. You can get this number from the endorsement stamp with which you stamp the members' checks when you process tithing (you can also contact the Church's 800-number).
* I went directly to the bank (as opposed to contacting our FM Group contact or our stake facilities coordinator first).
* I explained to the bank teller that we lost our drop box key and that we needed a replacement.
* I provided the bank teller with the Church's bank account number. If the teller asks you for the entity associated with this account number, it'll likely be "The Corporation of the First Presidency of the Church of Jesus Christ of Latter-day Saints" (or something along those lines; I can't totally remember the name that the bank teller shared with me) as opposed to "The Church of Jesus Christ of Latter-day Saints" or your ward's name.
* The teller checked the signature card to confirm that I was on it and then provided me with the key.
Your experience might vary, but for the most part it should be similar to what I outlined above.
Adding the terms deposit box and overnight and night bank box in case anyone searches by those terms.