WASHINGTON—American therapist Majd Kamalmaz was stopped at a government checkpoint in Damascus one night in February 2017 while visiting family in Syria’s fractured capital. He hasn’t been seen or heard from since.

For nearly two years, the Kamalmaz family quietly worked with the State Department and Federal Bureau of Investigation to try to find out what happened to the 61-year-old Virginia man who has spent decades treating victims of war around the world. But their efforts have turned up almost nothing. Relatives aren’t...