TOKYO— Nintendo Co. ’s biggest battle these days isn’t against other game makers. It is against companies such as Apple Inc. that are gobbling up the same parts Nintendo needs to make its hit Switch machine, people in the industry say.

Nintendo has told suppliers and assemblers it hopes to make nearly 20 million units of the Switch device in the year ending in March 2018, people involved in the discussions said. Though the company’s official sales target for the year is 10 million, strong demand suggests it can sell many more—if it can make them.

The problem is an industrywide capacity shortage for components used in smartphones, computer servers and other digital devices. These include the NAND flash-memory chips that store data, as well as liquid-crystal displays and the tiny motors that enable the Switch’s hand-held controllers to imitate the feel of an ice cube shaking in a glass.

“Demand for our NAND flash memory has been overwhelmingly greater than supply, and the situation is likely to stay for the rest of this year,” said a spokeswoman at Toshiba Corp., the industrial giant that is leaning on flash memory to survive. She cited demand from smartphone makers—Apple and Chinese companies are among Toshiba’s customers—and data centers.

People in the industry say the rapid expansion of web-based services for corporations has driven demand for computer servers that use flash memory. Continued demand for Apple’s iPhone 7 and a 10th anniversary model of the iPhone expected later this year are also keeping parts makers at full capacity, helping power Japan’s economy to its longest growth streak since 2006.