The White House announced yesterday that President Reagan intends to nominate Adm. William J. Crowe Jr. to succeed Gen. John W. Vessey Jr. as chairman of the Joint Chiefs of Staff, effective Oct. l.

Vessey, 63, who received a battlefield commission on the Anzio beachhead, is the last of the World War II veterans to serve on the Joint Chiefs of Staff. Crowe, 60, did not graduate from the U. S. Naval Academy until after World War II but served in Vietnam.

In his letter to Reagan requesting to be relieved of the chairmanship on Sept. 30, Vessey, who would have completed his second two-year term next June, wrote: "I ask to leave in September only because my wonderful wife has saluted and obeyed for the past 40 years.

"She cheerfully endured numerous family separations, countless short-notice moves, life in strange places . . . . I promised to have her in our home in Minnesota before the first snow of the 1985-86 winter, and I must keep my promise."

Reagan, in his response, expressed his "profound thanks and admiration for your dedicated service in bringing about a safer, more stable world . . . . Ever since that day 40 years ago when you stood as a first sergeant on a beach at Anzio, you have met and overcome all challenges and never looked back . . . . "