The answer is: He can. The 12th Amendment states that anybody who is eligible for the presidency under Article II of the Constitution (a natural-born citizen age 35 or older) is eligible for the vice presidency. Clinton is a natural-born citizen over 35, so he qualifies. The putative roadblock to a Clinton vice presidency--the 22nd Amendment--doesn't apply. This hastily worded and passed amendment, designed to block another multi-multi-term presidency such as FDR's, only bars the election of a president to more than two terms in that office. It doesn't prevent a two-term president from running for the vice presidency.
The 25th Amendment affords Clinton another route to the vice presidency: In the event the vice presidency is vacated, the president appoints a new veep, subject to confirmation by Congress. (This is how Gerald Ford and Nelson Rockefeller became vice president.)
Nothing in the Constitution would prevent Vice President Clinton from becoming president via succession. Finally, another scenario could return Clinton to the White House without a pit stop at the vice presidency. If both the presidency and the vice presidency were vacated and Bill Clinton were the speaker of the House, he would become president under the 1948 presidential succession act. (Next in line, the president pro tempore of the Senate and then the cabinet officers in the order specified by the act.)
Press Box thanks Yale Law School professor Akhil Reed Amar, American University Washington College of Law professor Jamin B. Raskin, George Washington University Law School professor Jeffrey Rosen, and Northern Illinois University professor of history David Kyvig.