Both are in common usage. The US Constitution uses the hyphen as well as capitalization: the Vice-President . Purists might say the hyphen is needed to avoid confusion with a possible president of...
The former vice-president, now the president, would nominate a new vice-president who has to be approved by both houses of Congress in order to take office.
There is no succession to the office of Vice President. Any President, no matter how he attains the office, can appoint a replacement Vice President to be confirmed by a majority of both houses of...