The now-defunct Florida company where Acting Attorney General Matthew Whitaker served as a paid advisory-board member once posted a promotional video featuring someone who has since risen to even greater heights in the administration: Donald Trump.

It is the latest twist in the story of World Patent Marketing Inc., which has attracted a sudden spotlight since Mr. Trump appointed Mr. Whitaker to lead the Justice Department last week after former Attorney General Jeff Sessions was ousted. The Trump Organization, which licenses...