The real story

The person who brought it actually died before his smallpox was identified, so while the film is correct that over 6 million people were vaccinated, there was no hunt for a carrier:

The N.Y.C. Health Dept. learned of the first cases Friday, Mar. 28 from Dr. Dorothea M. Tolle, medical superintendent of Willard Parker Hospital, a municipal institution fo treatment of contagious diseases. It was eventually discovered that the original carrier was Eugene La Bar, an American who had returned to U.S. from Mexico City where he had lived since 1940. He died at Willard Parker before doctors ascertained that he had smallpox. Altogether, 12 persons were stricken; 2 died. The Dept. of Health tracked down and sequestered several hundred people presumed to have been exposed to the disease. Later, it was suggested the entire population of the city be vaccinated. On Friday, May 2, Dr. Weinstein, then Commissioner of Health, announced the end of the outbreak. Within the space of 28 days a total of at least six million three hundred and fifty thousand people had been vaccinated in the city. Practically everyone in N.Y. was then immune.