The long-running inquiry reached a turning point this week when F.B.I. agents seized telephones and computers from Mr. Giuliani’s home and office in Manhattan, the people said. At least one of the warrants was seeking evidence related to Ms. Yovanovitch and her role as ambassador, the people said. In particular, the federal authorities were expected to scour the electronic devices for communications between Mr. Giuliani and Trump administration officials about the ambassador before she was recalled in April 2019, one of the people added. The warrant also sought his communications with Ukrainian officials who had butted heads with Ms. Yovanovitch, including some of the same people who at the time were helping Mr. Giuliani seek damaging information about President Biden, who was then a candidate, and his family, the people said. At issue for investigators is a key question: Did Mr. Giuliani go after Ms. Yovanovitch solely on behalf of Mr. Trump, who was his client at the time? Or was he also doing so on behalf of the Ukrainian officials, who wanted her removed for their own reasons?