Apple Inc.'s quest to adopt advanced displays for its next-generation iPhone hinges on a single supplier in the Japanese countryside.

Canon Tokki Corp., surrounded by rice fields in the city of Mitsuke in Niigata Prefecture, has a near monopoly on the machines capable of making screens with organic light-emitting diodes, which enable sharp, vibrant displays that use less energy. A unit of Canon Inc., the company of 343 employees has spent more than two decades perfecting the manufacturing equipment used by OLED screen makers.

But there's a problem: Canon Tokki has a growing backlog even after doubling output in 2016. The potential production bottleneck is raising questions over Apple's ability to feature OLED displays in next year's iPhones, and whether the Cupertino, California-based company will be able to line up additional suppliers. The current wait for a machine, which can cost more than ¥10 billion ($85 million) each, is about two years.