They looked all though the Constitution trying to find out if he could.
They must have overlooked the 12th Amendment. That's the one that ends with:
But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.
Now, if Bill Clinton is "ineligible to the office of President" ... and he is ... then he can't be Vice President.
If you're not clear on this, I can't help you. No one can.