This side quest can be started by visiting Hudson's clothing store (or rather what's left it). The store is located to the right of the Bank of the Prophet building.
Once you've entered the store go to a dark cellar and find the Vox cipher. It's on a shelf to the left of the sink.
You won't be able to continue this mission until you've released Lady Comstock's ghost and therefore began your search of the three tears. This is when you'll be allowed to go deeper into the bank and that's where the code is being stored.
After you've used the elevator to reach the main part of the bank you'll end up in front of a large vault door. Proceed to the left corridor and continue exploring this wing until you get to a bright room with a HOARDER sign on the wall.
Once Elizabeth has deciphered the code use the typewriter to open a secret passageway located behind a bookshelf. The hidden room contains a lot of money, a locksmith's set, Voxophone #69, Infusion #23 and a Burstgun rifle. There's also a smaller locksmith's set on a desk to the right of the typewriter.