Marine Corps service
Mueller was accepted for officer training in the United States Marine Corps in 1968, attending training at Parris Island, Officer Candidate School, Army Ranger School, and Army jump school.
In July 1968, he was sent to South Vietnam where he served as a rifle platoon leader with Second Platoon, H Company, 2nd Battalion, 4th Marines, 3rd Marine Division. In December 1968, he earned the Bronze Star with 'V' distinction for combat valor for rescuing a wounded Marine under enemy fire during an ambush that saw half of his platoon become casualties. In April 1969, he received an enemy gunshot wound in the thigh, recovered, and returned to lead his platoon until June 1969. For his service in and during the Vietnam War, his military decorations and awards include: the Bronze Star Medal with Combat "V", Purple Heart Medal, two Navy and Marine Corps Commendation Medals with Combat "V", Combat Action Ribbon, National Defense Service Medal, Vietnam Service Medal with three service stars, Republic of Vietnam Gallantry Cross, Republic of Vietnam Campaign Medal, and Parachutist Badge.
Mueller eventually became aide-de-camp to 3rd Marine Division's commanding general, then-Major General William K. Jones, where he “significantly contributed to the rapport” Jones had with other officers, according to one report.
Reflecting on his service in the Vietnam War, Mueller said “I consider myself exceptionally lucky to have made it out of Vietnam. There were many—many—who did not. And perhaps because I did survive Vietnam, I have always felt compelled to contribute.”