That only applies to President. A new Vice Presidential appointment would be voted on by both Houses of Congress.
With the office of the Senate President being vacant, the Senate President Pro Tempore would preside in his place.
There have also been several occasions throughout American history where a Vice-President vacated the office. Prior to the 25th (?) amendment, the office remained vacant until the following election. So I would think there would be several instances of precedent to look back to.