The text immediately before that spells out what appointments the president can make:

he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.

In context it’s clear that the vacancies being referred to are the vacancies of the positions the president has appointment power over, and that explicitly excludes senators since their method of appointment is already provided for by the Constitution.