Is there a bona fide religious exemption for a mandated vaccine? Not really.
Here is a study paper of the topic from the National Institute for Health. The mainstream, dominant world’s religions are decidedly for health and the prospering of human life. “I have come that they may have life, and have it to the full,” said Jesus (John 10:10), and Christianity has a long history of caring for the sick and founding hospitals. In premodern times, Buddhist monks often served as medical providers because Buddhism also seeks out health. The Qur’an states that Muslims have a duty to care for the sick, and Muslims were the first to scientifically investigate medicine and disease (after the Greeks). In Judaism, the Torah’s commandment to be fruitful and multiply implies health.
Some of the world’s religions object not to medicines and vaccines, but to their source material that they define as immoral. Muslims avoid vaccines that include gelatin from pigs or animals that were illicitly slaughtered (non-halal). Catholicism forbids vaccines rooted in cell lines derived from voluntary aborted fetuses. But these religions make allowances depending on the situation: there are no alternative solutions, to save the lives of children, for the greater good of the community and humanity.
The world’s dominant religious traditions are pro-health and pro-vaccine, with some notable exceptions. This is not surprising given the immense diversity of religions and religious experiences. Some Protestants have argued that vaccination interferes with God’s providential will, or that prayer and spiritual healing should be used instead of modern medicine; some non-Western religions have endorsed similar practices (for example, The Spirit Catches You and You Fall Down: A Hmong Child, Her American Doctors, and the Collision of Two Cultures, by Anne Fadiman). While there are always counter-examples, especially in the area of religious belief and practice, the point is that these sorts of anti-health religious practices are unusual. Religions have, for the most part, emphasized health and medicine.
Not discussed in the NIH paper is the complicated Christian apocalyptic fear of the mark of the beast (Revelation 13:17). This fear of an oppressive governmental control related to the end times dates from the 20th century, which imagined this mark of the beast to be a United Nations control system, the World Wide Web, Y2K, a tattoo, or an implanted biochip in the Left Behind series. This may be an overt or latent fear, but it is not uncommon in Christian cultures. The suspicion of evil coercive forces runs deep.
Some of the religiously identified anti-vaccination comes not from religion, but from culture. A 1988 Russian article led to massive vaccination hesitancy in Russia, despite the Russian Orthodox Church’s endorsement of vaccination. Vaccine hesitancy in Western cultures cites not religion but freedom of choice, autonomy over one’s body, and a fraudulent 1998 scientific paper. Yet religious groups can become infected with such beliefs, just as any group can. Ironically, it leads to a self-contradiction: religions that promote love of others, compassion, the good of the community, and self-sacrifice end up endorsing practices that harm the community when children and the immunocompromised get sick or die.
Ultimately, religions are complicated things. They instruct people what to eat and not eat, when to eat, what to wear, and how to live. They have led to violent crusades and to pacifism. It is not surprising that religions would interact with ideas about vaccination. What is surprising is during a worldwide catastrophe (4 million people dead, and more dying every day), and given the weight of the world’s religions being pro-vaccine during such a pandemic, we still imagine a religious objection to vaccines.