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 with politics. Mr. Luckey, it turns out, was put on leave, then fired, according to people familiar with...