Vaccines are approved if they are considered safe for people with long-term conditions. The approved vaccines are safe and effective for the vast majority of people; they have been tested on tens of thousands of people and assessed by experts. If you have a long-term condition you may be vaccinated sooner; your GP can advise on what cohort you are in.
A very small number of people who are at risk of COVID-19 cannot have the vaccine; this includes people with severe allergies. Everybody will be screened for potential allergic reactions before getting vaccinated. All vaccinators will have the training they need to deal with any rare cases of adverse reactions, and all venues will be equipped to care for people who need it, as with any other vaccine.