Larry Calo entered an elementary school in Glen Rock, N.J., for Sunday services of the Emergence Church, greeted fellow congregants and scanned the room for an open seat. Then he dipped his hand into a glass jar and pulled out a pair of orange PVC foam earplugs.

Before long, an electric guitarist, bass player, drummer and singer were tearing through a roughly 15-minute set featuring parish classics such as “King of My Heart.”

The...