MaJiG:
You will need to think about how you are going to change the code to get a key. After it is used once, the code to get key 14 will need to be changed.
How about an SD card shield? A small SD card with a single text file that has 14 codes in it. Every time you load the key box, you change the SD card to one with a new file on it. The file can be created with a text editor on a PC. The key pad could just be 4 push buttons labeled A, B, C, D and an enter button. If you use an voltage divider keyboard, it only requires a single analog pin. Codes could be something like 'AABDCADB'. The code file would then just look something like:
AABDCADB
DABCADBB
ABCDABCD
CDABASDD
...