Andra Blomkalns, chairwoman of emergency medicine at Stanford University, suspects that all 80 of the physicians who work in her Palo Alto, Calif., emergency department have been exposed to the novel coronavirus.

One who tested positive became very ill. At least a dozen others are awaiting test results. She wants to keep high-risk providers, like older physicians and one who had an organ transplant, from seeing patients in person. But she can’t lose too many of them because sick people are streaming into her department in...