The car could be in the Vatican car museum
Not in Vatican Museum.
there is a second museum for papal vehicles, not open to the public apparently.
I actually believe it IS in the Vatican Museum; a part in the Vatican Gardens that is not open to the public. I visited it several years ago, through the kind favor of a priest who was working in Rome. In this room is also the Jeep JPII was shot in, several other cars, and the coronation coach of Pius IX (I think it is PIus IX's...)