I think your question is really about how it was possible for young men in America to avoid military service in the Vietnam War, of which conscientious objection was only one means.

The United States practiced compulsory military service--"the draft"--from 1948 to 1973 (the "postwar draft period") under the Selective Service Act of 1948, which was extended and modified several times. The purpose of the draft was to provide adequate manpower for the military. It was neither the goal nor the practice to ensure that every capable young man served. In fact, during the draft period, the demographic situation ensured that there were many more young men available to be drafted than the military could usefully employ. Thus, it was never inevitable that one would be drafted. Men who were physically or mentally unfit were exempted; this included a host of seemingly minor medical conditions such as childhood asthma as well as homosexuality, which was regarded as disqualifying for medical service. The government also offered deferments for entire categories including, at times, high school and college students, married men, fathers, workers in essential jobs, and various "hardships" such as being the sole support of parents. Even for a man who was considered fit and did not have a deferment available, the chance of being drafted was, at many times, less than 50 percent.

It was also possible to avoid military service in Vietnam by joining a branch of the military that was unlikely to lead to deployment to Vietnam. The Air Force was the premier example of this: in fact, securing a safe spot in the Air Force was so popular that draftees were never required and the service could turn away many applicants. In fact, the other services accused the Air Force of manipulating the situation to ensure it had the pick of young men. It was also possible to directly enter the military reserve components and the National Guard, which were both usually "part time" commitments. Although some reserve and National Guard units and individuals were deployed to Vietnam, they were very few and those who actually went were effectively volunteers. Other uniformed services such as the Coast Guard, Coast & Geodetic Survey, and Public Health Service were exempt.

All these rules were subject to exploitation: a minor medical issue could be developed into unfitness, a young man might claim to have a psychological issue, a marriage might be hurried, and it's likely millions of young men attended college with draft avoidance in mind.

So what about conscientious objection, meaning that an otherwise fit young man had an objection to military service, primarily because of religious or similar objections to committing military violence? Conscientious objection was recognized in law, though the law developed significantly during the period. As the postwar draft period began, conscientious objection was closely identified with the so-called Peace Churches, small Protestant denominations that forbade violence and embraced pacifism, going so far as to counsel members not to fight even in self-defense. Examples include Amish, Quakers, Mennonites, Church of the Brethren, Jehovah's Witnesses, and Seventh-Day Adventists. The Peace Churches were pretty small, distinctive communities with demanding religious strictures, so there was little danger of young men who wished to avoid the draft falsely or opportunistically claiming to be members. Conscientious objection by bonafide members of Peace Churches was actually written into the Selective Service Act, by specifying that the law does not "require any person to be subject to combatant training and service in the armed forces of the United States who, by reason of religious training and belief, is conscientiously opposed to participation in war in any form" (emphasis added). The emphasized portion was generally understood to refer to the doctrines of the Peace Churches.

During the postwar draft period, however, individuals also claimed conscientious objector status based other criteria. Some said that they were pacifists and had a conscientious objection to violence and military service but on a basis other than a Peace Church's doctrines or theistic religion at all. In 1965, the U.S. Supreme Court ruled on a cased called United States v. Seeger in which the objector claimed that his objection to war was based on the writings of secular philosophers and had as a young man associated closely with Quakers but did not claim his objection was religious. The court said he should be treated as a conscientious objector because "the beliefs which prompted his objection occupy the same place in his life as the belief in a traditional deity holds in the lives of his friends, the Quakers." In Welsh v. United States, decided in 1970, the Court expanded this rule and said the draft law also had to respect "moral, ethical, or religious beliefs about what is right and wrong" that are "held with the strength of traditional religious convictions." Thus, Seeger and Walsh allowed nonreligious conscientious objection. This included conscientious objections that were members of non-pacifist religions; e.g., Catholicism and most Protestant denominations. If they personally had a moral objection to war, the fact their religious body approved of at least some wars did not undermine their position.

On the other hand, a decision from 1971, Gillette v. United States, upheld the government's position that a conscientious objector must object to participation in all wars, not only wars the objector considers unjust. So if you didn't want to fight in Vietnam because you considered it an illegitimate colonial struggle or had no quarrel with the Viet Cong, but you would have fought against Nazi Germany in WWII to save Jews from the Holocaust or would defend the United States against an invasion, you would not be accepted as a conscientious objector. This meant that members of non-pacifist religious that had a "just war" doctrine could not use that doctrine to avoid service.

A young man claiming conscientious objection was supposed to notify his local draft board at the time of registration or soon thereafter. He then filled out a form explaining his beliefs. This form remained in his file until he was actually called for induction. The local draft board then considered any other reason to defer service and only considered conscientious objection as a last resort. That consideration was also carried out by the local draft board. As suggested above, membership in a Peace Church, preferably since childhood, was the easiest to prove. If the local draft board did not agree, he would be ordered to report for induction, and if he refused to do so, he would be prosecuted, at which point he'd argue his case in court. This is what happened in the cases cited above.

Conscientious objectors were not simply returned to civilian life. They had the opportunity to perform noncombatant service within the military (e.g., serving as an unarmed medic or similar role). If this was not acceptable to their beliefs or if there was not such an opportunity, they could also be assigned to work in civilian alternative service. Some civilian alternative service jobs involved work on public lands and others were for religious organizations. The Peace Churches tended to operate their own programs that qualified.

In sum, there was a legal framework for conscientious objection during the Vietnam era, but it was not the primary means for avoiding military service, and it was still largely oriented toward religiously-motivated pacifists and in theory did not allow young men who simply objected to or did not want to participate in the Vietnam War only to enter.