I thought that was the case, but I seem to recall that there have been instances in which a person has been selected to fill a vacancy. Or is my memory just faulty here?
Senators can be appointed by their state’s governor. The House is governed by the Constitutional clause (Article I, Section 2: “When vacancies happen in the [House] Representation from any State, the Executive Authority thereof shall issue Writs of Election to fill such Vacancies.”)