A fast-growing digital currency that claims to be backed by U.S. dollars has become a cornerstone of the volatile cryptocurrency market. The problem: There isn’t hard evidence the cash supporting it exists.

Tether, whose main selling point is its tie to the U.S. dollar, has grown dramatically over the past year—its daily trading volume of around $3 billion trails only bitcoin’s $5 billion. Tether has also become a “crypto bank” for cryptocurrency businesses that have trouble maintaining real-world banking relationships, providing...