The U.S. has launched a criminal investigation into millions of fake comments filed to the Federal Communications Commission last year on rules governing the internet, according to an organization that has received a federal subpoena in the probe.

The subpoena from the Justice Department in Washington seeks all communications with the organization about comments it filed with the FCC regarding revision of Obama-era “net neutrality” rules, according to people briefed on the matter. The U.S. subpoena arrived just before the...