Neither the 22nd Amendment nor any piece of legislation offers an impediment to Biden staying on. Nor does history. George Clinton was Veep during Thomas Jefferson’s second term and James Madison’s first term. John C. Calhoun was John Quincy Adams’ VP, and stayed on during Andrew Jackson’s first term.
The harder question, of course, is political. Would it be an advantage to the Democratic presidential candidate to have Biden by her side? The answer lies in what happens over the next year and a half. If Obama winds up as unpopular as he was in 2014, there’d be no good argument for a third Biden term. However, President Obama’s job approval has bumped up about seven points in the last year. If that pattern continues, Obama will find himself at close to 60 percent by the time Democrats gather in convention, and a Biden vice presidency might well seem an attractive “third term” idea. And, unlike every VP who sought and won his party’s presidential nomination, Biden would be spared the burden of trying to prove himself “his own man.” He’d simply be the symbol of experience, free to distance himself from the less popular policies and ideas of the outgoing Chief Executive.