You can. It seems like a random event. It doesn't require a specific storyteller either.

Got my first looted vault about 2 years after game start, my main suggestion is that even if you can't carry much back with you, still go to the vaults and hack the broadcasting station thing to get the location of another sealed vault. The default spawn rate is pretty low so maintaining it via constant vault raids is the best way to make sure you can raid a vault when you are able to actually carry the loot back.