LAST LINE OF THE TEXT OF 12th AMENDMENT
"But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States."
Obama is not eligible to be president as he has served two terms (22nd amendment). Therefore, he is not eligible to be vice president.
Obama could be appointed to the office of VP. But he couldn't be elected to the office of VP. For example Gerald Ford was never elected as either POTUS or VP, he was APPOINTED to both. Barrack could be APPOINTED to either of those offices, but he couldn't be ELECTED to either. 22nd Amendment only restricts someone being ELECTED to the office if they've already served two terms. If Barrack were to be APPOINTED (as was Ford) then he could hold the office. So if Joe resigned right now and Kamala became POTUS she could APPOINT Barrack and then resign. Then Barrack would be APPOINTED to POTUS without having been ELECTED and the 22nd wouldn't come into play.
For 0bama to be appointed VP, both houses of CONgress would have to approve via a majority vote. Personally I don't believe there is a chance in hell that either team would vote for that. Reps wouldn't want him anyway, and Dems are theoretically smart enough to realize that would be tied up in every court for years until it reached SCOTUS. I could see half of the state AGs suing to stop it, individually. By the time a case reached SCOTUS the situation would be long over, and the only thing to be decided is what to do about the 12th, 20th, 22nd, and 25th amendments so nobody tried it again. That is - if SCOTUS doesn't just punt saying the point is moot.
What's worse is if Kamala is stupid enough to try it, she could also be stupid enough to step down before any court cases are filed - leaving the POTUS and VP both vacant. We would be left with the SoTH Mike Johnson as POTUS (currently a Rep) and the President Pro Tempore of the Senate Patty Murray (currently a Dem) as VP - and of course another open SoTH vote to replace Johnson.
Also that would mean for the first time ever, the Executive branch would be run by the top two ranking people in the Legislative branch just a day before.
It is a fun exercise, but there is zero chance of it happening.
Finally - Ford was not appointed to both offices. He was appointed and confirmed as VP, but he assumed the office of President (via succession and the 25th amendment) when Nixon stepped down. Immediately and Automatically.
I suppose if Kamala were to take the office of POTUS right now she could appoint BO (or anybody) as VP and then let Congress and the courts fight it out. While they're doing that she could appoint anyone at all as acting VP.
edit: I did some research and it looks like there can be no such thing as an "acting VP".