It is the nation’s largest appeals court, covering nine Western states and dealing with a staggering set of topics from social questions like same-sex marriage to border issues to land resource matters. Because of its size, experts say that Mr. Trump would be unable to reverse its ideological makeup even if he were able to fill all eight vacancies. Some of those nominees would replace judges who had been appointed by other Republican presidents.

But there is no dispute that Mr. Trump has the chance to push it to the right. The dynamics of the court could change in many subtle ways — producing, for example, more sharp dissents that catch the attention of the Supreme Court, said Leonard Leo, the executive vice president of the Federalist Society.

Plus, it is hard to measure the effect of the loss of Judge Reinhardt, who was seen as a major influence on the liberal wing of the court and a talented and articulate legal protector of liberal views. “The death of Judge Reinhardt means more than the loss of a liberal vote,” said Arthur Hellman, a law professor at the University of Pittsburgh and a leading expert on the appeals court.

The mounting vacancies throw the future of the Ninth Circuit into the continuing Senate clash over the federal judiciary, one area where Mr. Trump has had success, with the enthusiastic assistance of Senator Mitch McConnell, the Kentucky Republican and majority leader. Since Mr. Trump took office, the Senate has confirmed not only Justice Neil M. Gorsuch of the Supreme Court, but also a record 14 appeals court judges and 14 district court judges.

Many more are in the pipeline. “I believe that’s the most important thing we are doing,” Mr. McConnell told a newspaper editorial board last week in Kentucky.

The Trump administration has already put forward two nominees for Ninth Circuit openings. One, Mark Bennett, the former attorney general of Hawaii, has the support of the state’s two Democratic senators.