It is extremely unlikely your child will experience serious side effects that could cause a long-term health problem after getting a COVID-19 vaccine.
Long-term side effects following any vaccination are extremely rare. In the past, vaccine monitoring has shown that if side effects are going to happen, they tend to happen within six weeks of receiving a vaccine dose.
For this reason, a portion of the children in each age group in Pfizer’s clinical studies were followed for at least two months following vaccination, and no serious vaccine-related health problems were reported. Plus, there is a safety monitoring plan in place specifically for children and adolescents given the COVID vaccine.
https://healthychildren.org/English/health-issues/conditions/COVID-19/Pages/The-Science-Behind-the-COVID-19-Vaccine-Parent-FAQs.aspx (American Academy of Pediatrics)