Four years ago, one of China’s largest agricultural importers sent representatives to the Democratic National Convention in Charlotte, N.C., hoping that meetings with elite party officials might yield business opportunities. The company, the Dandong Port Group, was particularly focused on the governors in attendance, according to an interview with Dandong’s general counsel broadcast by Chinese state television. But now, the company’s widening influence is coming under scrutiny by federal prosecutors, who are examining the relationship between Dandong’s wealthy and connected chairman, Wang Wenliang, and Gov. Terry McAuliffe of Virginia, a Democrat who was elected in 2013. A federal law enforcement...