My guess (and this is only a guess) is that if the institution is physically outside of the State of Vatican City --- that is, operating in Italy --- Italy law apparently has some kind of jurisdiction.
If foreign currencies are involved, I believe international regulators can intervene as well.
I imagine that international financial transactions can spark very complicated jurisdictional disputes.
But what do I know. Any smart FReepers here who know more, who can comment?
I know the Lateran Treaty of 1929 governs relations between the governments of Italy and the Vatican State, and that Italy has certain rights - but I do not know the details.
Perhaps, but it occurs to me that Italian “oversight” never seemed to slow down the Mafia.