Suppose that it is 2007 and Dick Cheney is Vice President. He resigns to become a senator from Wyoming. What steps would be taken to fill this vacancy?
This is spelled out in the 25th Amendment, Section 2:
Whenever there is a vacancy in the office of the Vice President, the President shall nominate a Vice President who shall take office upon confirmation by a majority vote of both Houses of Congress.
And that's really all there is to it. It's pretty much the same as any other "advice and consent" position, except that the House also has to approve it, not just the Senate.