The Securities and Exchange Commission has raised an uproar about what might seem to be a basic legal question: During an investigation, does the SEC have the power to subpoena personal emails or files from internet companies such as Google, Yahoo or Dropbox?

The Email Privacy Act, a bill unanimously passed by the House of Representatives in April, would explicitly deprive the SEC (and other civil agencies) of the ability to obtain personal emails from internet companies. It would also close a loophole allowing agencies like...