OK, it eventually worked. Even though I had beat the game from beginning to end with two controllers in local play with online set to "no online", I said to myself just one more time, and I played again with 2 controllers and the vault item unlocked on the 20th level. Why it was the 20th I have no idea...
I posted this rather than editing my original post to nothing just in case this info could be helpful to someone. So I'd say, if it didn't work after beating the game once, restart it from the end with the start button and try again? It's short anyway and I just kept going to the door, without barely even attacking, not even opening chests for keys because I always had the door key in my "inventory". Unlimited continues!