Ethan Zhang needed to get back to work. Work was in Ivory Coast, however, and since January the global coronavirus outbreak had stranded the 26-year-old translator in mainland China.

Then friends told Mr. Zhang of a way he could get his hands on what might be the world’s most coveted prize: a coronavirus vaccine. Though China’s vaccine candidates have not formally been proved safe or effective, officials have been injecting them into thousands of people across the country, ostensibly under an emergency-use policy. One such campaign, his friends said, was underway in the city of Yiwu in eastern China.