President Trump’s promise to investigate voter fraud has drawn predictable responses from Democrats and the media, who insist there is no such thing and have been fighting for years to prevent any inquiry into the matter. But an investigation in Mr. Trump’s hometown shows that the problem is real.

In 2013 the New York City Department of Investigation—the storied law-enforcement arm of city government, which houses and manages all the city’s inspectors general and investigators—decided to test the system. City investigators...