Constitutional FAQ Answer #20
Q20. "How old must the VP be? If the VP is less than 35 and the President dies and the VP is sworn in as President, who determines if he is eligible for the office?"
A. The age of the Vice President is not specified in the original version of the Constitution, but the method of selecting the Vice President was clear: the VP was the person who came in second. So, to come in second you had to have been running for President, and you therefore needed to be at least 35.
Today, the rule is set by the 12th Amendment, which states that no person who is ineligible to be President may be Vice President.