A major plank of Elon Musk’s plan for expanding X into an “everything app” has been grafting a payments network onto its main social networking service. The details of this project have remained blurry, but dozens of documents the company has submitted to state regulators—which have not previously been made public—give a new view of the business. Those plans include a Venmo-like payments feature that will let users store money on their X accounts, pay other users or businesses, and even buy good...