Just months before the spectacular fall of WeWork, one of the country’s most gilded startups, chief executive Adam Neumann summoned the heads of the New York Stock Exchange and Nasdaq to one of his homes in the Hamptons.

WeWork was going public and each executive wanted Mr. Neumann to list on their exchange. In return, he wanted their support for a cause he had championed—environmental sustainability—and asked them to ban meat or single-use plastic products in their cafeterias, according to people familiar with the matter....