WASHINGTON—A veteran Republican operative and opposition researcher solicited and raised at least $100,000 from donors as part of an effort to obtain what he believed to be emails stolen from Hillary Clinton, activities that remain of intense interest to federal investigators working for special counsel Robert Mueller’s office and on Capitol Hill.

Peter W. Smith, an Illinois businessman with a long history of involvement in GOP politics, sought and collected the funds from at least four wealthy donors as part of the plan to...