Maybe.
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," & the 22nd says that you can only be elected to the presidency twice (& if you succeed to the presidency & serve for more than half of one term, then that counts as one of your terms), so the question would come down to the Supreme Court deciding whether "constitutionally ineligible to the office" includes a prohibition on being "elected to the office more than twice."
A very strict reading of the Constitution would allow it, because the 22nd Amendment only says that a person can't be elected President more than twice, without saying anything about whether or not a person can serve more than twice.
So, if the Supreme Court were to take into account the words exactly as they're written, then the answer to your question is yes. If they take the clear intent of the 22nd Amendment into account, then the answer is probably not.
I mostly agree with this. But an important point you didn't touch on - isn't there legal prevention of a president serving longer than 10 years? Let's say a Biden/Obama ticket won in 2020. If Biden died and Obama took over as POTUS, he'd be removed from his office exactly two years after doing so and the sitting VP would ascend to the presidency.
No, there is no 10 year limit. For instance, it is still possible for a person to serve as President for more than 10 years, simply by being elected Vice-President multiple times with Presidents who do not complete their terms. No matter how many times that happened or how many years, that person would still be eligible to be elected to one term of their own.
However, as Obama has been elected twice to the Presidency, he is ineligible to be elected President, and thus arguably ineligible to the Vice-Presidency. However even under this strict interpretation of how the 22nd Amendment interacts with the 12th Amendment, under current law, if Obama were to become Speaker of the House. and the Vice Presidency were vacant, he could then succeed to the Presidency upon that office becoming vacant.
