A veteran Republican activist whose quest to obtain Hillary Clinton’s emails from hackers dominated the final months of his life struck up a professional relationship with Lt. Gen. Michael Flynn, the former national security adviser to President Trump, as early as 2015, and told associates during the presidential campaign that he was using the retired general’s connections to help him on the email project.

The late Peter W. Smith, an Illinois financier with a long history in Republican politics, met with Mr. Flynn in 2015,...