Facebook Inc. executive and virtual-reality wunderkind Palmer Luckey was a rising star of Silicon Valley when, at the height of the 2016 presidential contest, he donated $10,000 to an anti-Hillary Clinton group.

His donation sparked a backlash from his colleagues. Six months later, he was out. Neither Facebook nor Mr. Luckey has ever said why he left the social-media giant. When testifying before Congress about data privacy earlier this year, Facebook Chief Executive Mark Zuckerberg denied the departure had anything to do...