The main impact of the Vietnam War on immigration to the United States had to do with an increase in immigration from Vietnam and the surrounding areas.
During the Vietnam War, the United States, of course, was supporting an anti-communist government in South Vietnam. It also supported anti-communist movements in neighboring countries such as Laos. In 1973, the US pulled out of the war and, in 1975, North Vietnam achieved final victory over the South. As this happened, large numbers of South Vietnamese began to flee the country. The first wave of these refugees were government officials and others who had reason to believe that they would be persecuted by the communists when they won. The US felt a responsibility to these people and admitted more than 100,000 of them to the US. Before this, there had been essentially no Vietnamese in the US.
Within a few years, another wave of Vietnamese immigration occurred. These were the “boat people” who were desperate enough to leave Vietnam that they took to sea in small, overcrowded boats. In the 20 years from 1981, the US accepted over half a million refugees from Vietnam. Many were boat people while others were the children of American servicemen and Vietnamese women. In addition to these immigrants, people from other countries who had fought against communist regimes fled to the US. These include people like the Hmong from Laos. These refugees were also connected to the Vietnam War.
In these ways, the Vietnam War led to the influx of a large number of Southeast Asian immigrants to the US.