The CDC and FDA have multiple surveillance systems. People are encouraged to submit information to the Vaccine Adverse Event Reporting System, which predates this pandemic. Experts comb through the submissions to identify and follow-up on adverse reactions that might be due to the vaccine.
The CDC also debuted a smartphone-based tool called v-safe for COVID-19 vaccines. If you register for the program, you will be asked via text message how you are feeling every day for a week after vaccination and less frequently after that. If your answers indicate a potential problem, someone will contact you for more information.