The airport says a lot about Cortez: The single-engine planes that fly into its one-room airport seat nine passengers at most. The city of about 9,000 is known largely as a gateway to beautiful places like Mesa Verde National Park and the Four Corners Monument. But COVID-19 vaccines have made Cortez a destination in its own right.

“We had a couple fly in to get their vaccine from Denver that couldn’t get it in the Denver metro area,” said Marc Meyer, director of pharmacy services and infect...