Under pressure from regulators, laboratory firm Theranos Inc. has stopped collecting tiny vials of blood drawn from finger pricks for all but one of its tests, according to a person familiar with the matter, backing away from a method the company has touted as it rose to become one of Silicon Valley’s hottest startups.

The move is a setback to the Palo Alto, Calif., company’s ambition to revolutionize the blood-testing industry. As a result of the halt, Theranos is operating more like a traditional lab that draws blood with...