KIEV, Ukraine—Joe Biden’s involvement with Ukraine and his son’s work in the country, which President Trump has repeatedly said should be investigated, began when the former vice president was serving as the Obama administration’s point man on relations with Ukraine and rooting out bureaucratic corruption.

Soon after Moscow invaded the country in 2014 and annexed its peninsula of Crimea, Mr. Biden took a lead role in U.S. efforts to support the fragile government in Kiev. He shuttled back and forth to the Ukrainian capital,...