He'd have a hard time making that case. The 12th Amendment says that no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States. It also says that the President and Vice President cannot be from the same state. As husband and wife the Clintons would have a hard time establishing that.
Fair enough. He could obviously carpetbag to another state. He would be constitutionally ineligible to be VP if there were more than two years left in the term. Clearly, he is ineligible to run again for a full term. He is constitutionally eligible to serve up to two more years, thus he is eligible to serve two more as VP. I think that he could actually serve two years at the end of her first term after the VP mysteriously met his demise, then serve two years at the end of her second term when that VP mysteriously met his demise.
A party could nominate two people from the same state if they were willing to write off the electoral votes from that state for the VP candidate, and if someone was appointed to the Vice Presidency the residency rule would not be an obstacle.