We do not currently know of any synonyms for Vietnam War.
The proper noun Vietnam War is defined as:
The war that occurred in Vietnam, Laos, and Cambodia between 1955 and 1975 between South Vietnam (backed by the United States and SEATO) and North Vietnam aiding the South Vietnamese communist guerilla army known as the Vietcong (supported by the People's Republic of China and the Soviet Union) for the struggle for total Communist control of Vietnam as one state.