The core question is – all vaccines by now were tested on a very large number of people, but they were not tested over long period of time. So, then the question is: during long-term testing of vaccines in last, say, 40 years: is it often that a vaccine is seen as safe after 1 year, but problems are discovered after a longer time period. Note, this is not about complications discovered simply by testing more people, but specifically 3-5 years after a specific patient got vaccinated.
I did a very brief search of vaccine trial reports several months ago and did not find any. Medicines seems to be more prone to side effects over time, as they are taken continuously. Maybe, others can find examples, and when we will have a baseline: say, 5 out of 50 vaccines had a side effect found after 3 years and how severe they typically are.