Reading this article I have to wonder what "religion" all these SF police employees belong to that denies them access to the vaccine.
Do they have to declare which religion they're a part of and then open up the Bible to the page that says "thou shalt not inject thyself with mRNA vaccines"?
The First Amendment essentially prohibits the establishment of a list of approved vs. not-approved religions. There is a large body of First Amendment case law that relates to claimed religious beliefs, and the restriction that the government cannot prohibit a person from exercising their religious beliefs. The courts therefore avoid bright-line answers to the question.
You can look at Friedman v. Southern Cal. Permanente for an example where a court found that a belief is not a religious belief, finding that veganism is not a "religious creed" within the meaning of the California Fair Employment and Housing Act. Plaintiff was told to get a mumps vaccination (which involves chicken embryo) as a condition of employment, refused, and the offer of employment was refused. The law, §12940a makes it unlawful "because of the religious creed of any person, to refuse to hire or employ the person...". Subdivision (l) says that
Religious belief or observance, as used in this section, includes, but is not limited to, observance of a Sabbath or other religious holy day or days, reasonable time necessary for travel prior and subsequent to a religious observance, and religious dress practice and religious grooming practice as described in subdivision (q) of Section 12926.
§12296(q) further contributes a definition of religious terms:
“Religious creed,” “religion,” “religious observance,” “religious belief,” and “creed” include all aspects of religious belief, observance, and practice, including religious dress and grooming practices. “Religious dress practice” shall be construed broadly to include the wearing or carrying of religious clothing, head or face coverings, jewelry, artifacts, and any other item that is part of an individual observing a religious creed. “Religious grooming practice” shall be construed broadly to include all forms of head, facial, and body hair that are part of an individual observing a religious creed.
The Fair Employment and Housing Commission then created a regulation California Code of Regulations, title 2, section 7293.1 that defines "religious creed"
‘Religious creed’ includes any traditionally recognized religion as well as beliefs, observations, or practices which an individual sincerely holds and which occupy in his or her life a place of importance parallel to that of traditionally recognized religions.
The court notes that
religious creed extends beyond traditionally recognized religions to encompass beliefs, observations, or practices occupying a parallel place of importance “to that of traditionally recognized religions” in an individual's life.
The court then points to the leading cases from the Supreme Court related to deciding what a religion is: US v. Seeger, 380 U.S. 163 and Welsh v. US, 398 U.S. 333. And the court points out that government agencies are granted wide latitude in interpreting enabling legislation (under which the government might allow or disallow an action).
It us noted that as of 2002, there were no California cases deciding what constitutes a religious creed w.r.t. FEHA and that regulation. The court then recited various statements from other California cases regarding the characterization of a religion – the main relevant point is that a religion does not have to be theistic to be a "religion". The court also reviews federal employment discrimination laws, and again considers the difference between traditional and non-traditional religions. There are many snippets in the review of the law section saying things such as that
the court should find beliefs to be a religion if they ‘occupy the same place in the life of the [individual] as an orthodox belief in God holds in the life of one clearly qualified.’
We can now move to the question in §6, Is Veganism a Religious Creed For Purposes of the FEHA. This court states that
The test we apply is that set forth in Judge Adams's concurring opinion in Malnak which has been adopted by the Third, Eighth, Ninth, and Tenth Circuits
which they say
presents the best objective method for answering the question whether a belief plays the role of a religion and functions as such in an individual's life
At this point, we can't reasonably guess how some individual would object on religious grounds to the vaccination mandate, but we do at least know what aspects of the law would be relevant and where the "rules" are laid down. Until someone actually alleges that they have a religious belief that prohibits vaccination, we can't analyze those arguments. The gist of the "veganism is not a religion" is it is too narrow a belief to constitute a religious belief. The court found that did not find that it is a "belief system (which) addresses fundamental or ultimate questions", and that it does not a address fundamental questions such as "the meaning of human existence; the purpose of life; theories of humankind's nature or its place in the universe; matters of human life and death; or the exercise of faith".
The article you link to details some of that
The city’s policy states that employees must fill out a form and “answer all relevant questions regarding the religious belief, practice or observance that prevents them from getting the COVID-19 vaccination.” They may also be required to submit additional information such as letters from a religious leader, articles from religious scholars or excerpts from religious or sacred texts.
The department said it would review requests for accommodation on a case-by-case basis.
It also notes that employees may be reassigned different jobs with less public interaction, even if an exception is granted.
FYI, in the US there are some religious groups that broadly object to various forms of medical treatment. Whether they are suitable to be employed in the police, where they e.g. may have to administer CPR or call an ambulance is an interesting discussion. It would depend on what kind of medical treatment they object to, I guess. I don't know if vaccination of any kind is usually mandatory for SF police.