However, there is not yet any evidence I've seen that the vaccine can cause Bell's Palsy. There's not conclusive evidence it can't, but the FDA says: "The observed frequency of reported Bell’s palsy in the vaccine group is consistent with the expected background rate in the general population."
https://www.snopes.com/fact-check/covid-vaccine-bells-palsy/
(Please do correct me if I'm out of date and there's anything showing statistical significance, but I don't see anything new from a quick Google.)
When you vaccinate a lot of people, some of them will develop medical problems that would have developed regardless of the vaccine. It's not impossible you will be proven right once there is more data, but until then it is spreading misinformation to say that the vaccine did cause the problem.
The traditional way (as used by the Oxford/Astrazenica covid vaccine, and others) is to show the body what the virus looks like by giving it non-harmful bits of protein that look like the bit of the virus the immune system will need to recognise.
The mRNA ones (like Pfizer's or Moderna's) use mRNA code to tell your body to create those harmless proteins, and then the rest is the same - your immune system learns to recognise them, which is good because it will be more likely to recognise the real virus.
The mRNA technique is just moving the "create these harmless things" from the lab to inside the body. It doesn't change how your body fights the virus compared to traditional vaccine methods.
(mRNA vaccines have also been undergoing safety tests for years, these ones for covid are the first approved but not because previous/ongoing ones were shown to be unsafe.)
e.g. info from CDC: https://www.cdc.gov/coronavirus/2019-ncov/vaccines/different...