No, you can get a COVID-19 vaccine and other vaccines at the same visit. Your body should develop protection the same as it would whether you get the COVID-19 vaccine alone or with other vaccines.
Previously, the CDC recommended waiting at least 14 days between getting the COVID-19 vaccine and any other vaccine out of an abundance of caution in monitoring the effects of the COVID-19 vaccine. It was not due to any safety concerns. We now know that it is safe for COVID-19 vaccines and other vaccines to be given at the same visit.