Nelson Rockefeller
Richard Nixon appointed Gerald Ford to the VP position after Spiro Agnew resigned. This was the first VP vacancy that occurred after the ratification of the 25th amendment which gave the president the power to appoint a new VP if the office became vacant.
Federal judges are appointed. The President nominates a candidate for a vacancy on the bench, and the Senate votes whether to approve or reject the nomination.
If you appointed and were not ELECTED by popular vote then you can not truly run for RE-election to the same position.
In the event of a vacancy in the presidency the vice president shall step in and perform the duties of the president. If the vice president is unable to hold this position then the next person in line would be the Speaker of the House of Representatives.
The electoral college that elects the president also elects the vice president. In the event that a new vice president is appointed because of a vacancy, both houses of Congress have to confirm the president's appointee.
If the presidency is vacancy. the VP becomes president and and new VP is promptly appointed . If the president and VP both die at once, the Speaker of the House becomes president.
amendment 25
This happened twice - in 1973 when VP Agnew resigned and Gerald Ford was appointed VP and again in 1974 when Ford became president and Nelson Rockefeller was appointed VP. ( Before that a vacant vice--presidency remained vacant until the next election. )
If the office of vice-president becomes vacant , the president nominates someone to fill the position and both houses of Congress must confirm the nomination to make it official.This law appears in amendment 25, clause two: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.
The President.
The US constitution makes the vice-president the immediate successor of the president and the 25th amendment provides a procedure for filling a vacancy in the position of vice-president. The US Congress established a succession list for emergency cases.