In conversations with members of his circle and with others he tried to recruit to help him, the GOP operative, Peter W. Smith, implied he was working with retired Lt. Gen. Mike Flynn, at the time a senior adviser to then-candidate Donald Trump.

“He said, ‘I’m talking to Michael Flynn about this—if you find anything, can you let me know?’” said Eric York, a computer-security expert from Atlanta who searched hacker forums on Mr. Smith’s behalf for people who might have access to the emails.

Emails written by Mr. Smith and one of his associates show that his small group considered Mr. Flynn and his consulting company, Flynn Intel Group, to be allies in their quest.