Mike Howell was looking at his phone in an airport lounge this month when he saw a letter from the attorney general of Missouri questioning whether President Joseph R. Biden Jr. had the “mental capacity” to sign the pardons and executive orders he issued during his final months in office. Mr. Howell, executive director of the Oversight Project, a branch of the conservative Heritage Foundation, sensed an opportunity, he said in an interview. For months, he had been comparing Mr. Biden’s signature on dozens of official documents and noting that many of them appeared identical. Before boarding his flight, Mr....