I will try here to sketch an answer starting from where @MattDMo left off.
How do we evaluate possible long-term side effects of a vaccine? (With specific application to the recently authorized vaccines for SARS-CoV-2).
Phase IV clinical trials
As @MattDMo has pointed out, the definitive answer can be given only in the Phase IV of the clinical ...
You are correct that we have been creating and approving vaccines for a very long time, so the procedures for looking at efficacy and side effects of vaccines in general long-term are quite well developed. They occur via so-called "Phase IV" trials, also known as post-marketing surveillance.
However, one complicating factor with at least the Pfizer/...