People who are breastfeeding can get vaccinated. The American College of Obstetrics and Gynecology (https://www.acog.org/clinical/clinical-guidance/practice-advisory/articles/2020/12/vaccinating-pregnant-and-lactating-patients-against-covid-19) recommends that the COVID-19 vaccine be offered to lactating people who are eligible.
Clinical trial data was not collected on people who were breastfeeding. However, there is no known risk to breastfeeding infants. No ingredients in the COVID-19 vaccines are known to be harmful to the safety of breastmilk. Antibodies may pass to the baby through breastmilk and offer some protection, but this has not yet been studied. If you have questions about getting vaccinated, a discussion with your healthcare provider might help you make an informed decision.