Special Counsel Robert Mueller has requested that Cambridge Analytica, a data firm that worked for President Donald Trump’s campaign, turn over documents as part of its investigation into Russian interference in the 2016 U.S. election, according to people familiar with the matter.

Mr. Mueller asked the firm in the fall to turn over the emails of any Cambridge Analytica employees who worked on the Trump campaign, in a sign that the special counsel is probing the Trump campaign’s data operation.

The...