My wife had a severe reaction to the MMR shot as a child in the late '70s. One in a million. She remembers scores of doctors coming into to look at her. She hasn't had anything more than a sore arm to anything since. Our pediatricians were kind enough to alter the vaccination schedule for our kids to give the MMR in isolation, but they had no issues at all.
I am not a doctor, and this is only an anecdote
From what I understand with regards to most flu vaccines it's an egg allergy that is the most common problem and the CDC has some guidelines on that. (https://www.cdc.gov/flu/prevent/egg-allergies.htm) Still, 1 in 1.3 million have anaphylaxis after receiving a flu shot which means well over 100 cases of it happening a year. Just as with the lottery, someone has to be the lucky winner. I don't believe egg is used in the COVID-19 vaccine, but something else is causing those to have trouble to have trouble.
As per standard drug disclaimers in commercials, if you are allergic to the COVID-19 vaccine, please don't take the COVID-19 vaccine.