A native of Minneapolis, Vessey was a 16-year-old Roosevelt High School student in 1939 when he began his 46-year military career by falsely claiming to be 18 so he could join the Minnesota Army National Guard's 59th Field Artillery Brigade, a unit of the 34th Infantry Division. His unit was activated for World War II, and he took part in combat in the North African and Italian Campaigns. Vessey received a battlefield commission to second lieutenant during the Battle of Anzio, and served as a field artillery forward observer until the end of the war. He remained on active duty after the war, and continued to advance through positions of increasing rank and responsibility, including assignments in West Germany during the Cold War. He graduated from the University of Maryland University College in 1963, and received a Master of Science degree from George Washington University in 1965. In addition, Vessey completed the United States Army Command and General Staff College and Armed Forces Staff College courses and the program of study at the Industrial College of the Armed Forces. During the Vietnam War, he served as executive officer of the 25th Infantry Division Artillery and acting commander of 2nd Battalion, 77th Field Artillery, and he received the Distinguished Service Cross for heroism during the Battle of Suoi Tre. He served as commander of the 3rd Armored Division Artillery from 1967 to 1969, and the division chief of staff from 1969 to 1970.