I have largely sidestepped the issue of COVID-19 and religious liberty; let me be clear: I opposed Draconian restrictions we saw on religious services in the early phases of the pandemic, which in my mind were clearly unconstitutional. Yet, with respect to my Catholic background, prelates went out of their way to be sensitive to civil leadership; my senior citizen Mom, who often attended (non-obligatory) daily mass, had to settle for televised Sunday mass, no eucharist.
When it comes to vaccines, however, there are some nuances. At the risk of oversimplification, there have 2 principal objections to vaccination: stem cell use in certain vaccine development and matters of conscience. Most people may not be familiar with the former. The use of stem cells from aborted preborn children is considered an abomination. However, note this is not a factor for the 2 leading mRNA vaccines which dominate the market Pfizer and Moderna. The second is more complicated, and I'll not review here in detail; let's simply point out this is not a blank check on human behaviors. For example, you have to consider it's not just your own health you risk but you could be morally responsible for passing the virus on to people with serious health issues. Still, by the linked source, over 80% of Catholics support COVID-19 vaccination and only about 7% oppose.
In fact, Catholics may be the highest vaccinated religious group (I've also seen high numbers for Jews) with 82% at least partially vaccinated 2 months ago. The Church hierarchy has been unambiguously behind vaccination (see here and here). Every member of my nuclear family has been fully vaccinated since mid-May. I don't know about all my 21 nephews and nieces, but several I know about have been. I took lots of shots as a kid as an Air Force brat, who lived in multiple states and in Europe. I never heard or read anything from the Church suggesting vaccines were morally suspect; they are fully consistent with the Church's consistent pro-life perspective, of responsibility to oneself and to others.
Of course, Christianity extends beyond Catholicism, and the evidence I've seen suggests general vaccine acceptance (significantly less than among Catholics, with the possible exception of less-tolerant white evangelicals, although scant evidence of doctrinal restrictions. If there was a more principled religious basis, I thought it might be Christian Scientists who often prefer praying to conventional medical protocols, but the cited source suggests they could accept civil mandates.
So why some "conservatives" and libertarians are outraged that SCOTUS hasn't been that sensitive yet to religious exemptions for things like vaccine mandates, there really aren't any substantive doctrinal reasons from the perspective of organized religions, but the courts take a liberal position on what constitutes religion; I suppose SCOTUS would accept the Anti-Vaxxer Church of God, if someone started it tomorrow, arguing rubbish like mRNA vaccines are the Mark of the Beast.
Over the weekend, GML's Nate Thurston interviewed Jeff Myers, reportedly a philosophy PhD and president of Summit Ministries ) I think episode 576).. Both Thurston and Myers take a dim view of things like vaccine mandates. Myers made a comment (I don't have a transcript in front of me) to the effect that breakthrough and unvaccinated are comparably infectious. It really depends on the specifics of what is being stated. There have been some studies that viral loads are comparable among the two infected groups, at least for the highly infectious Delta variant, and it is thought that high load is related to transmission. However, there's not a lot of hard data on breakthrough transmissions to others. In general, the vaccinated are far less likely to get infected; if and when breakthroughs happen, they tend to be milder and less serious, and they're generally less likely to transmit a virus to others (this is the rationale for the cocooning strategy) . (For example, I recall an early Wuhan study where they looked at close contacts of the reinfected and found no evidence of propagated infection.) Here's a relevant CDC comment:
For the Delta variant, early data indicate vaccinated and unvaccinated persons infected with Delta have similar levels of viral RNA and culturable virus detected, indicating that some vaccinated people infected with the Delta variant of SARS-CoV-2 may be able to transmit the virus to others.(163, 164, 177-180) However, other studies have shown a more rapid decline in viral RNA and culturable virus in fully vaccinated people (96, 177, 180-182). One study observed that Delta infection in fully vaccinated persons was associated with significantly less transmission to contacts than persons who were unvaccinated or partially vaccinated.(181)
I think in part inconsistencies in state reporting has led the CDC to focus on hospitalization and deaths among breakthrough and unvaccinated. But to give one example (Virginia), the unvaccinated are 6 times more likely to get infected than the vaccinated and 2.5 times more likely than the partially vaccinated. The fully vaccinated: <1% chance of infection, <.03% of getting hospitalized, <.01% chance of dying.