Categorically No.

Thou shalt not take the name of the LORD thy God in vain; for the LORD will not hold him guiltless that taketh his name in vain. (Exodus 20:7)

Promises made to others in God's name engage the divine honor, fidelity, truthfulness, and authority. They must be respected in justice. To be unfaithful to them is to misuse God's name and in some way to make God out to be a liar. (1 John 1:10)

The Catechism of the Catholic Church teaches that it is a duty to reject false oaths that others might try to impose; an oath may be made false because it attests to a lie, because an illegitimate authority is requiring it, or because the purpose of the oath is contrary to God's law or human dignity. Wikipedia

No

Only those things which are prohibited by the Church are prohibited by the Church-- or, put another way, the Church teaches that being fully the sort of Catholic the Church teaches us to be is how to be Catholic, or yet another way, the Church teaches what it teaches, which includes that what it teaches is logically consistent, or another way "Truth does not contradict Truth".

Catholicism is also a public religion-- that is to say, it's not gnostic or otherwise dependent on secret truths known only to elites. There is no apocrypha in the ancient sense, no secret gospels too powerful for the minds of "lesser" men. All of the Church's moral teachings are, as a matter of principle, publicly available, even if they may not always be easily understood, and a bishop refusing to teach someone moral principles in order to keep them secret has something of the flavor of simony to it.

That said, laws hardly ever deal with real things, because they would then be practically unenforceable in the sort of neutral, objective, non-discriminatory way most modern societies want law enforcement to work. A law, for example, might prohibit eating a certain food, gathering in a certain place, or not wearing a certain item of clothing. The Church does not categorically require or permit any of that. However, the Church does categorically forbid sin and categorically extoll virtue and all of those describe both virtuous and wicked actions. The Church, by and large, leaves determining when an action is pernicious and when it is morally obligatory to the prudential judgment of the person making the decision whether or not to take that action. The Church provides advice and teaching and guidelines to help with that-- see e.g https://christianity.stackexchange.com/a/78754/17529 -- but ultimately it is up to the individual practitioner in all but the most exceptional cases to determine what God requires of them and then do it, and in those rare cases where it is really obvious that someone is wrong about that, it's the local pastor who has the primary competence to correct and teach his congregation, not any big centralized body in the Vatican or anything. So, while theoretically only that which the Church forbids is, of course, forbidden by it, practically someone would have to be messing up really, really bad for their specific religious objection to actually not hold up on the grounds that "Catholicism doesn't prohibit that", since almost all actions or non-actions could be a sin and the Church has a lot of respect for the importance of experiential knowledge, prayer, and prudential judgement.

This is a legitimate question, but the answer cannot be generalized as the question wishes it, because each case is different (see points of consideration #1, #2 and #4 below).

Sample Case

For the sake of this question let's consider a specific case: a hypothetical regulation for compulsory Covid-19 vaccination:

• On January 1st, 2021 a USA federal regulation goes into effect, mandating Covid-19 vaccination of every resident of USA, free of charge, by March 31st, 2021
• using a vaccine of one's choice as long as it passes Stage 3 Clinical trial
• even though 1) there are published potential minor side effects, 2) there may be long term adverse side effects which are currently unknown due to accelerated nature of this 1st generation vaccine research.
• There is widespread announcement in all media channels a few months before. The text of the regulation as well as the disclosure of the vaccine research and production are also made available in the federal government websites.
• Exemptions: 1) certain groups of people whose health conditions made them susceptible to the known side effects, 2) you're pregnant, 3) you have a doctor's note
• The reason for the regulation is for making the community as a whole safer (herd immunity) when return to normalcy is desired (for economic and other community related reasons)
• There is at least one vaccine that is produced ethically according to Catholic moral principles (example: not derived from abhorrent origins such as cells from non-implanted or aborted foetuses)

A Catholic is understandably faced with a decision that calls for risking his/her health when obeying the regulation.

Is the regulation ITSELF legitimate?

Let's review Aquinas's Treatise on Law on the legitimacy of a human law. It needs to fulfill 4 criteria:

1. Ordination of reason (i.e. follow natural law principles)
2. For the common good (i.e. not corrupt, or end up injuring the common good)
3. By one who has care for the community (i.e. by justly elected / appointed official having jurisdiction)
4. Promulgated (i.e. published through a channel that everyone has access to; no surprises, no secrets)

From the details above, the regulation appears legitimate.

Points of consideration

Now we consider the Catholic individual's act as he/she decides whether to be vaccinated or to refuse. Let's consider 2 grounds for refusal:

• Safety: Let's say there are credible articles by credible people that the insufficiently tested first generation vaccine may have unforeseen long term side effect that can damage someone's health permanently, since normal vaccine development usually takes 10-15 years. So this Catholic is considering postponing vaccination until the 2nd or 3rd generation which would not have been developed by the deadline of March 31st, 2021.
• Ethical: Let's say all of the available vaccines as of Mar 2021 have some assocation one way or another with non-implanted or aborted foetuses in the past, either in research only or also in production. For a background, see the document Moral Reflections on Vaccines Prepared From Cells Derived from Aborted Human Foetuses attached in a 2005 letter response from the Pontifical Academy of Life, especially the 3 distinctions: 1) formal vs. material cooperation, 2) immediate vs. remote, 3) active vs. passive cooperation. Furthermore, due to the developing situation, there are yet-to-be-resolved differences in opinion between US Bishops and UK Bishops on whether to comply with a vaccination regulation. So an individual Catholic has to make a judgment call to his/her situation.

Here are some points of consideration from the Catholic perspective:

1. Whether the act of refusal itself is licit under the Catholic natural law principle? An act is wrong if either of the means or the telos / ends of the act is wrong/evil. In this instance the telos (the chosen aim of this act) is NOT to break the regulation, but for the preservation of one's health. Therefore the aim is OK, especially because one is still open to vaccination in the future (when it becomes safer, or when an ethical vaccine choice becomes available). The means of this act is the refusal itself, but the breaking of the regulation is merely "accidental" to the aim (like borrowing a friend's car to drive oneself to the hospital in an emergency without notifying your friend because there is no time, is not considered stealing). So this act is licit in the eyes of the Catholic church. For more guidance, see a lecture series from the Thomistic Institute Aquinas on Human Action and Virtue especially lectures 1, 2, 5, and 6.

2. Whether the regulation itself permits exemptions? Most regulations have exemptions that may not be religious in nature. But the OP question specifically asks for religious exemption, presumably because this is the only avenue to be exempted from the regulation. Therefore, for this hypothetical regulation, let's say the individual does NOT qualify under those exemptions, so the individual would like to say to the authorities that he/she doesn't want to be vaccinated for "religious reason", which can be worded as: "After taking all facts under consideration my moral conscience says that I should not take the 1st generation vaccine and my church allows me to break a regulation that is potentially injurious to my health. I appeal to the Free Exercise Clause." This is even though the Catholic Church does not prohibit you to be vaccinated, but let's say another church X does.

3. Whether the way US constitution phrases the Free Exercise Clause you have to belong to the institutionalized religion (church X), or whether the clause simply refers to your conscience?

   • If it is the former, you may have to fight harder, but can cite church X as a precedent and hopes the court can rule in your favor
   • If it is the latter, then the OP question is probably a moot point. But we still need to consider points #4 and #5 below

4. Whether the religious exemption for this regulation has been decided by the Supreme Court? Glancing through this article covering Supreme Court decisions over many types of Free Exercise Clause lawsuits, it's very clear that this is a case by case basis. With respect to Covid-19 related regulations, they are still untested in the courts. Therefore it seems one can legitimately test the court by disobeying the regulation anyway over conscience without appearing to blatantly ignoring the law, until it is tested in the Supreme Court. Once it's tested though, from the Catholic law-abiding principle it could be a more serious disobedience because the legitimate law criteria #3 (issued by legitimate authority) becomes much stronger.

5. Whether the Catholic church allows one to disobey a regulation that has been tested by the Supreme Court? We assume that one has made a justifiable personal decision regarding a specific case based on the Catholic natural law principle AND the Catholic church is silent BUT it is against a secular regulation already tested by the highest court in the land. Are there previous cases where the Catholic Church has allowed this implicitly or explicitly? If it is a Yes, then we may have a stronger reason to disobey licitly, even though we should not take it as a blanket statement to disobey all Supreme Court rulings simply over moral conscience.

Conclusion

If there is already a Supreme Court decision about the particular case, then criteria #3 (legitimate authority) becomes a lot stronger. But even so, if we can decrease the strengh of criteria #1 (ordination of reason) by appealing to either personal danger or the unavailability of ethical vaccine (described in 2 grounds of refusal above), it is probably licit, especially if the Catholic church is relatively silent (so far) on this specific regulation, leaving a faithful free to follow his/her conscience.

At any rate, when disobeying this regulation because one believes criteria #1 of the regulation is not sufficient (for example: it should not have been compulsory, or no ethical vaccine is available), one has to take the precautions so the purpose of the regulation itself is protected (criteria #2: community safety). In lieu of declining vaccination in March 2021 one can make a stronger case when appealing to the Free Exercise Clause in the Federal Court by demonstrating that one wears mask, maintains safe distance, performs self-isolation when exhibiting symptoms, and remains open for vaccination, and thus proving that the refusal to obey the regulation is done in good conscience.

UPDATE: The Congregation for the Doctrine of the Faith issued a 21 Dec 2020 Note on the morality of using some anti-Covid-19 vaccines that provides clarity on:

• moral reasoning for benefiting from the ethically compromised cell lines while strongly asserting that it is NOT a moral endors*m*nt
• moral imperative for organizations (pharmaceutical, governmental, etc.) to sponsor fully ethical vaccine development
• balancing the availability of fully / partially ethical vaccines with personal duty for health and common good
• that vaccination "must be voluntary" while needing to preserve personal health and common good:

  At the same time, practical reason makes evident that vaccination is not, as a rule, a moral obligation and that, therefore, it must be voluntary. In any case, from the ethical point of view, the morality of vaccination depends not only on the duty to protect one's own health, but also on the duty to pursue the common good. In the absence of other means to stop or even prevent the epidemic, the common good may recommend vaccination, especially to protect the weakest and most exposed. Those who, however, for reasons of conscience, refuse vaccines produced with cell lines from aborted fetuses, must do their utmost to avoid, by other prophylactic means and appropriate behavior, becoming vehicles for the transmission of the infectious agent. In particular, they must avoid any risk to the health of those who cannot be vaccinated for medical or other reasons, and who are the most vulnerable.