There is no "vault" on Counterfeit Island.
The Dragon Face Lock
At night, secretly enter the museum by opening the Dragon Face lock in the Underground.
The Black Widow's Lair
The Key Card you recovered from the scooter chase will open the locked barrier.
(see the two related questions)
The art museum on counterfeit island opens after you go to the guy at the dock.
You don't. It is just a distraction.
get all the ripped papers
You go to the city docks and enter it from there
Counterfeit IslandNo, Counterfeit Island is real. In fact, it'd be better named as "Art Island".
The art thief is back on Counterfeit Island. As part of Counterfeit island, you don't need to do anything in Early Poptropica except get the key from the curator at the Pop Art Museum, then return to Counterfeit Island to open the Inspector's house.
there is no mansion on counterfeit island
You do not die on Counterfeit Island.
the chest is just a distraction
There is no camera as an item on Counterfeit Island.
Go back to Counterfeit Island and use it to open the Inspector's house in the Country Side. On the way, the street artist gives you a clue as to what to look for in the house.
You will have to swim for it : the lighthouse is on Big Nate Island, not Counterfeit Island.