Chris Christie, the former New Jersey governor, on Thursday became the latest Republican politician to be fooled into making a campaign video on behalf of a Democrat. Mr. Christie is one of many of Mr. Trump’s current and former associates available for hire on Cameo, an app that allows users to commission personalized videos from minor — and increasingly major — celebrities. The video, which cost $200, was framed as a jovial message to a person named Greg, who Mr. Christie was prompted to encourage to return to New Jersey, Greg’s former home. What Mr. Christie did not know was...