Were there attempts to make this argument say, during, the Vietnam War era, or otherwise?

Yes, during WW1, just after the Selective Draft Law was passed in 1917 (precursor to the 1967 Selective Service Law) several were charged and convicted of refusing to register and Emma Golding, the anarchist and prominent Socialists were also convicted of encouraging others to refuse to register. After the trials, SCOTUS, snatched the three cases from the appeals courts and heard them immediately. Together, Arver v US, (245 U.S. 366), Goldman v US, and Ruthenberg v US, make up the "Selective Draft Law Cases") The NY Civil Liberties Union (NYCLU) took up similar arguments in 1969, challenging the 1967 updated law along the same lines and arguing the Selective Draft Law Cases were incorrectly decided, specifically arguing 1) the Military Selective Service Act of 1967 is unconstitutional since it exceeds the powers granted to the federal government, and 2)that historical evidence demonstrates that the framers of the Constitution did not intend to grant Congress the power to conscript.

See: Leon Friedman, Conscription and the Constitution: The Original Understanding, 67Mich. L. Rev.1493 (1969)

The facts from Friedman and the NYCLU:

Joseph F. Arver and Otto H.Wangerin refused to register and were indicted on June 8, 1917, tried the following month before a United States district court in Minnesota, found guilty, and sentenced to one year in prison. The Supreme Court granted a writ of error directly to the trial court and argument was presented on December 13 and 14, 1917, along with the cases of other draft resisters from New York. At the same time the Court heard the appeals of Alexander Berkman and Emma Goldman,"two noted anarchist leaders who had been found guilty of conspiring to counsel resistance to the draft law in New York, and the appeals of Charles E. Ruthenberg, Alfred Wagenknecht, and Charles Baker, prominent Ohio Socialists who were convicted of encouraging a young man not to register.

In asserting the invalidity of the draft, the defendants pressed two primary arguments:[1] that the thirteenth amendment's prohibition of involuntary servitude deprived Congress of any power to conscript; and [2] that the draft conflicted with the militia clauses of the Constitution since the federal government had effectively destroyed the state forces by drawing all the members of the state militia into federal service and shipping them overseas. In the course of their argument, the defendants traced the history of English military organization, emphasizing that no general conscription law had been passed in England prior to the twentieth century.They also claimed that the acts and regulations of the draft unlawfully delegated legislative authority to the President.

Chief Justice Edward White authored the opinion rejecting all those arguments and upholding the law. Characterizing the power to conscript as "an essential attribute of sovereignty." He cited the large number of nations enforcing compulsory military service in 1917, concluding: "It would be a contradiction in terms to declare the Government of the United States a sovereign, endowed with all the powers necessary for its existence, yet lacking in the most essential of all-the power of self-defense."' C.J. White also cited the many colonial and state laws in force before 1787-almost 200 were listed-calling for compulsory militia service by all male citizens and noted the federal draft was proposed (although not passed) in 1814 and the fact that a conscription law was enacted during the Civil War showed the practical exercise of the power and was therefore a recognition of it. The court further rejected the relevance of the militia clauses of the Constitution, since men were taken directly into a federal army by the1917 law rather than as members of a federalized state militia. The Court dismissed the thirteenth amendment argument by pointing out that the sole purpose of the amendment was to abolish chattel slavery, not to eliminate compulsory governmental service.

The Arver court wanted to settle the selective service challenges completely and for all time, and as much as Friedman and the NYCLU disagreed, 13A is and was only about dissolving the badges and incidents of slavery.

Slavery and involuntary servitude were enforced, both prior to and after the adoption of the 13th amendment to the Constitution of the United States, through widespread public and private violence directed at persons because of their race, color, or ancestry, or perceived race, color, or ancestry. Accordingly, eliminating racially motivated violence is an important means of eliminating, to the extent possible, the badges, incidents, and relics of slavery and involuntary servitude. (National Defense Authorization Act for Fiscal Year 2010, Pub. L. No. 111-84, § 4707, 123 Stat. 2190, 2838–39. Division E of the Act is denominated as the Matthew Shepard and James Byrd, Jr. Hate Crimes Prevention Act.)

The final sentence of 13A is "Congress shall have power to enforce this article by appropriate legislation." Congress created the Civil Rights Act(s) under this power, as well as the Hate Crimes Act in the quote above. But the mandate is limited to laws necessary and proper to eradicating 'badges of slavery', nothing more. Involuntary Servitude includes 'peonage', a kind of wage-slavery common in sharecropper agreements, company stores, and arrangements where an employer extends credit to an employee and works for free to repay it but never can. Petty criminal charges and excessive fines and convict labor has also been judged as peonage and a badge of slavery (see 1903 Alabama Peonage Trials) in certain circumstances. I'm in favor of a looser understanding of peonage to bring student loans under 'badges of slavery', but courts won't buy that anymore than military service for which a person receives wages. Non-compete employment contracts were struck down as involuntary servitude by courts until about the 1970s, but that's the furthest they've taken it outside traditional civil rights.

Conscription isn't in the constitution, but was discussed in the Federalist Papers as a means of common defense, and because 'militia' in 2A meant all males, able bodied and 14 years old, the founders, the constitution itself and the Supreme Court are comfortable with the legality of conscription as a sensible means of homeland defense.

The Militia Act of 1791, ancestor of the present National Guard system, mandated universal obligatory service at the state level, with state forces available to the federal government in time of specific emergency. All else was to be handled by the standing army or by volunteer forces raised for a clear and limited purpose.

Vietnam, as a foreign war, and depending on your point of view, an immoral war, was missing that connection to homeland defense, so there's an argument that the obligation of common defense doesn't attach to foreign wars, but that's a losing legal argument. Conscientious objector status is available and has been recognized for religious objections since the founding (Quakers, etc) but the moral objections are limited to moral objections to all war, not merely foreign wars or unpopular wars, but all. Gillette v US, 401 U.S. 437 (1971) With an all volunteer military- despite the taxing of National Guard forces in Vietnam and more recently- the chances of a draft and conscription are very remote, but not involuntary servitude as 13A precisely defines it.