Amazon.com Inc. on Thursday launched an unusual plan to establish a second corporate headquarters in North America, signaling the online retail giant’s growth ambitions for the years ahead and setting off a race among local and state governments hoping to win it over.

Amazon said it was soliciting proposals for a new campus location, which eventually could result in the company investing more than $5 billion and creating up to 50,000 high-paying jobs, many in software development and most of them new. Proposals are due Oct....