Last time we were at the Boardwalk Inn, the safe in our room didn't work when we got in our room. It was Sunday afternoon, and the person they sent up to look at it couldn't get it to work and said they would have to call the company that fixes the safes on Monday. Anyhow, we were told that there were safe deposit boxes down at the front desk that we could use if we liked. So they do exist but I don't know if that is true at all resorts, just deluxe resorts....etc.
Edited: just found info in the following blog. Refer to paragraph after bell services photo, it addresses safety deposit boxes:
http://blog.wdwinfo.com/2013/03/12/staying-safe-at-walt-disney-world-tips-for-you-your-family/