Adverse events are monitored continuously as long as the patient is still a part of the trial (which often extends past approval).

Once the product is approved, the FDA has an adverse event monitoring system that collects reports and analyzes them for signals that might imply a previously-unknown adverse event.

Of course it take more than just one case to for a new signal to be identified (unless it's really obvious like the patient dies right after getting the vaccine), so sometimes it may be a year or more until there is enough data to say "yes, we know nearly all the risks associated with this vaccine".