If one were to do a search for "Bell’s palsy and SARS-CoV-2 vaccines" one would find several more recent studies that show far lower risk than the one earlier study that was posted. One study speaks directly about the early study and says they overestimated the risk. Other later studies show no difference. Amazing that the study with the highest rate was the one that was posted.
Earlier study
https://www.thelancet.com/journals/laninf/article/PIIS1473-3099(21)00076-1/fulltext
Therefore, the observed incidence of Bell's palsy in the vaccine arms is between 3·5-times and 7-times higher than
would be expected in the general population
Study about the study above
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8184125/
Here, we offer a different interpretation of their findings and statistical consideration of risks associated with mRNA and non-mRNA SARS-CoV-2 vaccines.
Given this, and considering Bell's palsy as the possible outcome of individual doses, the observed incidence in the mRNA vaccine trials would be roughly 1·5 to three times higher than in the general population (table).
https://jamanetwork.com/journals/jamainternalmedicine/fullarticle/2779389
When compared with other viral vaccines, mRNA COVID-19 vaccines did not display a signal of facial paralysis
https://jamanetwork.com/journals/jamaotolaryngology/fullarticle/2781367
Association of COVID-19 Vaccination and Facial Nerve Palsy
Conclusions and Relevance In this case-control analysis, no association was found between recent vaccination with the BNT162b2 vaccine and risk of facial nerve palsy.
If we were to do a search of Bell's palsy and flu vaccine we can find similar differences in various studies. Some show no difference. A few show a higher incident rate, though more research needed.
There are also cases of Bell's palsy that have been associated with the disease itself.
I would not be surprised if the vaccines increase risk of Bell's palsy when compared to an unvaccinated group. I would also not be surprised if Bell's palsy in vaccinated groups is similar in rate to disease infected people.