WASHINGTON—A longtime Republican activist who led an operation hoping to obtain Hillary Clinton emails from hackers listed senior members of the Trump campaign, including some who now serve as top aides in the White House, in a recruitment document for his effort.

The activist, Peter W. Smith, named the officials in a section of the document marked “Trump Campaign.” The document was dated Sept. 7, 2016. That was around the time Mr. Smith said he started his search for 33,000 emails Mrs. Clinton deleted from the private server...