The storage should be available right from the start. As soon as you're done with the introductory quests and Karen's in the lobby during the day, she has an option saying "Check in with Karen". After a short conversation the storage is unlocked and you can enter whenever you like.i can't get the last clue on the let's split up gang quest because the storage is still locked for me.
If you've advanced some other stories without ever having that conversation and the option's not available anymore, it's possible you've managed to sequence break something.