And they see tremendous opportunity in the fact that Mr. Trump is the first Republican president whose nominees can be confirmed by simple-majority votes, especially since he is likely to fill an unusually large number of vacancies. Mr. Trump started with 21 open appellate seats because after Republicans gained control of the Senate in 2015, they essentially shut down the confirmation process. Six additional appellate judgeships have opened since his inauguration, and nearly half of the 150 active appeals court judges are eligible to take senior status — semiretirement that permits a successor’s appointment — or will soon reach that age, according to Russell Wheeler, a Brookings Institution scholar.

As a result, Mr. Trump is poised to bring the conservative legal movement, which took shape in the 1980s in reaction to decades of liberal rulings on issues like the rights of criminal suspects and of women who want abortions, to a new peak of influence over American law and society.

“What makes this a unique opportunity in modern history is the sheer number of vacancies, the number of potential vacancies because of the aging bench, and the existence of a president who really cares about this issue in his gut,” said Leonard A. Leo, an informal adviser to Mr. Trump on courts who is the executive vice president of the Federalist Society.