After a 40-year tenure that has witnessed unprecedented innovation and meteoric growth in the uniform and industrial services sector and related markets, Cintas Corp. Chairman Scott Farmer now sees his company’s biggest long-term challenge in figuring out how to compete not only against other laundries, but with online companies such as Amazon and retailers such as Wal-Mart that he sees as eyeing the linen, uniform and facility services and direct-delivery market.

“We converted the no-programmers all along,” says Farmer, who after 18 years as CEO retired from that post on June 1 to make way for former Executive Vice President ...