Service Education Reallocation for Vacant Elected offices Act of 2021 or the SERVE Act of 2021 This bill modifies the authority to make nominations for cadets or midshipmen to the military service academies in situations where a Member of Congress has not submitted a nomination due to death, resignation from office, or expulsion from office and no successor member has been sworn in to office. Specifically, in the event a Senator does not make a nomination for one of the specified reasons, the nominations must be made by the other Senator from the state concerned. In the event a Representative does not make a nomination for one of the specified reasons, the Senators from the state of the congressional district concerned must make the nomination, with nominations divided equally among Senators and any remainder going to the senior Senator from the state.