ROME—When the Vatican opened two tombs near St. Peter’s Basilica on Thursday morning, there was cautious optimism that it might shed some light on a mystery that has haunted Italy, and the Catholic Church, for more than three decades.

Emanuela Orlandi disappeared from a city street at the age of 15 in 1983, and her fate has become one of Rome’s longest-running whodunits, with her family unrelenting in their quest to uncover the truth. A tip that her remains might be buried in the Vatican City’s Teutonic Cemetery prompted officials...