The question of under whose authority the Avengers fall isn’t exactly hotly debated. I mean, in-universe it is, chiefly by middling, conniving, “Hail Hydra”-whispering politicians, but not really amongst their fans, be they real or imaginary. By nature these powered people are extra-governmental, at best work in tandem with the U.S. government, at worst openly opposing it. Many of the Avengers could and have been designated as enemy combatants, while a few might not even be classified as “alive.” Which actually raises a more interesting question:
Do The Avengers Have Rights?
As a collective non-profit organization, the Avengers don’t have many legal rights. It’s by far easiest to describe the group by its individual members.
Tony Stark, for example, has the most rights of any team member. Being a U.S. citizen and a civilian, Tony actually has the same basic human rights as many of the people he saves. Tony is easy. Realistically, being incorporated (and a billionaire) gives him more political influence than any other member of the team. He would be entitled to vote, and would likely vote libertarian if he remembered to fill out the card Pepper gave him. Pepper, for her part, examines all issues very carefully and votes in the best interests of the people as she sees it, usually leaning Democrat in national elections.
Bruce Banner is likewise technically an American citizen, but as a hunted fugitive effectively has nothing but basic human rights. It is important to note that Banner has never been tried or convicted of any crimes by any civilian or military court, and so retains his right to vote in presidential elections (and primaries, if he be party affiliated). However, it’s a safe bet a tactical squad would be at any registration offices within several minutes of his arrival. And with no known current address, I doubt he’d be able to stand the interminable line wait anyway.
Steve Rogers, Captain America, is active military, so he really only has as many rights as the American government says he has. He was also legally dead after an operation during World War II, so that usually puts a damper on being alive. If S.H.I.E.L.D. hadn’t done it already, after the Battle of New York the reinstatement of Steve’s citizenship and being-alive-status were most likely fast-tracked. Social Security number, back-pay and hazard pay, disability, even probate would have to have been straightened out. Steve remembers voting for FDR, and would be likely to vote Republican for their appeal to “greater” times. Until he heard the candidates speak. Then he would then listen to Democrats speak and, horrified, vote for an independent candidate who had no chance of winning but seemed nice. Yep, Steve Rogers believes in Bernie Sanders.
Clint Barton, Hawkeye, is in a similar predicament to Cap, being a covert wetworks operative. His existence alone should barely be recognized by the federal government for fear of reprisal for sanctioning his illegal activities, presumably including some light assassination here and there. Though he could legally vote, he would not, viewing his rotating cabal of spymasters as exactly that, irrespective of political affiliation.
Natalia “Natasha” Romanoff, the Black Widow, has actually fewer rights than Clint or Steve, being a foreign national. Defection likely landed her at least an American citizenship if not full amnesty for her prior crimes, however this citizenship could then easily be revoked following the dissolution of S.H.I.E.L.D., possibly even threatening her with deportation (if they physically could manage it). This would leave her vulnerable to reprisal from foes and past allies no longer kept at bay by the weight of her employment by the U.S. intelligence network. Ironically, her covert identities, being created by the federal government for official purposes, likely carry the weight of legitimate citizenship as long as they remain on the books, and so may grant her more liberties than her actual identity. If she ever voted, it would likely only be done ironically and as part of her cover, or for a strong female candidate.
Thor Odinson is a long-lived, super-strong alien, and thus doesn’t actually have human rights. He may, however, retain some form of diplomatic immunity as the son of foreign sovereign. He has also been recognized as an ally of S.H.I.E.L.D. and therefor (tentatively) the American government. Though the notion of voting for one’s ruler might seem alien to him, given the legal opportunity to do so he would likely vote for Ted Nugent. He would not understand why this was funny.
Pietro and Wanda Maximoff, Quicksilver and the Scarlet Witch, basically have the same rights as Natasha, being foreign-born. Both would be labeled as enemy combatants or terrorists by the U.N., Wanda was likely absolved of her part in these crimes after switching sides and helping to save the population of Sokovia and the world at large. Her affiliation with the Avengers likely protected her from sanctions by the American government directly following the events of Age of Ultron. Until she is naturalized and takes her citizenship oath (which she wouldn’t), she cannot vote.
The Vision is technically an artificial life form, manufactured and the intellectual and physical property of Ultron, itself an artificial intelligence partly developed and therefor owned by Tony Stark and Stark Industries. If Stark classifies his Iron Man armor as a high-tech prosthesis, legally speaking, Vision is a really really fancy stapler. Vision accurately predicts the winner of every election ever, but refuses to tell anyone of the outcomes.
The Avengers are sadly a good representative sample group for what makes a citizen-superhero in the Marvel Cinematic Universe. Their current and eventual compatriots are likewise … problematic, from a legal perspective.
Scott Lang, the Ant-Man is a convicted felon. Scott has basic American rights, but loses many personal freedoms including voting. He also can be denied work, entry into foreign nations, as well as access to public housing and–depending on state regulations–firearms. He can also never serve jury duty so … net win?
Dr. Henry “Hank” Pym, former Ant-Man, is part-way between Tony Stark and Clint Barton. His above-top-secret espionage and wetworks missions during the Cold War would classify him as a war criminal in parts of the former Soviet Union, but earned him a tidy salary and scientific funding. Hank was probably ready to vote for humanitarian and Nobel Peace Prize decliner Alexander Pierce up until he turned out to be the current head of HYDRA.
Sam “Falcon” Wilson is retired U.S. military. Technically he has never done anything wrong and retains all civilian rights and his work with the Avengers is merely “contract work.” He can, however, be recalled by the government to active duty if they so choose. Sam votes Democrat both on social issues and more recently on military spending.
James “Rhodey” Rhodes, a.k.a. War Machine, a.k.a. Iron Patriot, is active U.S. Air Force. They can do whatever they want to him. Rhodey votes democrat.
James Buchanan “Bucky” Barnes, the Winter Soldier, technically doesn’t exist. He was declared M.I.A. and presumed killed in 1945 and has since spent most of his life frozen and/or brainwashed by secret elements of HYDRA and the U.S.S.R. The concept of voting still confuses Buck.
Peter Quill, “Star-Lord” likewise was presumed missing in 1988 and would have been legally declared dead several years later. He is the only ( part-)Terran among the Guardians of the Galaxy, and thus the only one even theoretically allowed to vote. Peter would vote for Bart Simpson and think this was still incredibly funny.
The Agents of S.H.I.E.L.D. don’t vote. Most of the team was declared hunted criminals following the events of The Winter Soldier. Though relations have eased a bit and they have even managed to work with the U.S. government since, they are technically still unaffiliated. Their legal status is murky at best. Agents not tarnished by these events are either guilty-by-association or British. Sometimes both. In either event they know voting to be a farce.
Nick Fury doesn’t technically exist, and those who knew of him think him dead. An idealist at heart, Nick Fury secretly voted for Ralph Nader, Ron Paul, and is expected to vote Bernie Sanders in 2016.
 If they were a for profit corporation, it’s likely they would control the known multiverse, CEOed by his honorable chairmanness Victor von Doom.
 During the “Civil War” comics storyline, Tony was actually installed as the recognized Director of S.H.I.E.L.D., circumventing any official chain of command.
 Banner is also probably a little too disillusioned with the two-party system to bother with voting. The impulse however to say that Hulk is “Green Party” is … “sizable.”
 I’d imagine a new SSN would be issued, attached to his original birth certificate and army records. Whether Rogers was owed any compensation for his time spent on ice is arguable since it was wartime and his physiology not only ensured his survival but also left no lasting physical damage. Any personal civilian effects would likely have been snowballed with his military gear for “study.” Any released materials would have ended up with agent Peggy Carter, as Rogers had no living family and his best friend had recently fallen off a mountain. These saved mementos probably account for a large portion of the original pieces seen in his apartment during The Winter Soldier.
 This begins an interesting discussion on who qualifies for “human” rights. A better terminology to use in an age where human, near-human, and non-human sentient lifeforms are recognized as existing might be “universal” or “sentient rights.”
 It’s actually important to note that Barack Obama was likely never elected in the MCU. Since at least 2011, the standing POTUS has been Matthew Ellis, implying an original election date of 2008 and thus supplanting Obama as the nation’s 44th President.