Amazon.com Inc. will pay its employees to quit and help them start their own local package-delivery businesses, as the e-commerce giant competes for delivery drivers in the tightest U.S. labor market in 50 years.

Transportation operators, from long-haul truckers to retail chains like Walmart Inc. to startups such as Deliv, are increasingly competing for people to help them carry the millions of packages that are moving each day around the country. Demand for so-called last-mile delivery is booming as more Americans shop online....