WASHINGTON—U.S. spy agencies, already challenged in their missions relating to foreign-election meddling and North Korea’s weapons programs, and by a U.S. president who sometimes dismisses their work, face a new obstacle: the coronavirus.

Thousands of personnel at major intelligence agencies, including the Central Intelligence Agency and the National Security Agency, which conducts electronic eavesdropping, are working in staggered shifts. In any given week, as much as half of their workforce is sent home while the other half...