Drugs and vaccines are two very different things. Very different. Drugs work on different pathways and are often recognized by your body as "poison", even while being effective treatments.
Vaccines will either work to create immunity or they won't. Many vaccines will cause swelling or redness at the injection site. Every vaccine carries a universal warning:
"People sometimes faint after medical procedures, including vaccination. Tell your provider if you feel dizzy or have vision changes or ringing in the ears. As with any medicine, there is a very remote chance of a vaccine causing a severe allergic reaction, other serious injury, or death."
That's pretty generic stuff, and honestly could be applied to peanuts as easily as a shot.
Also, the flu vaccine (which I suspect the COVID vaccine will be similar to) comes in Live Vaccine and Inactivated Vaccine flavors. Live vaccines are generally regarded as safe and tend to be more effective. Live vaccines contain deactivated viruses which cannot give you the disease that you are trying to build immunity to.
You can read the side-effects of most major vaccines here:
https://www.cdc.gov/vaccines/vac-gen/side-effects.htm
