Are Orthodox Jews permitted to go into the Sistine Chapel in order to see the art? It is unclear to me if the Sistine Chapel is a "church" in current use and therefore prohibited, or if its use only for conclaves makes it OK?

Shalom, Jews are not allowed to enter a church that is used for worship - though some Rabbis permit entering a church no longer in use in order to visit it as a museum (see Aseh Lecha Rav, vol 4, 53). As far as I can ascertain, the Sistine Chapel is still used for worship - "The conclave [of the Sistine Chapel] also provides for the cardinals a space in which they can hear mass" (Wikipedia - The Sistine Chapel). See also the work "Tsaydah LeDerech" 78,9 where he writes expressly "visiting the churches in the Vatican is forbidden ... even the inner courtyard where the Popes' sermons are heard is forbidden to visit, because the sermon itself involves clear idol worship, and the courtyard is set aside permanently for this use". Blessings.