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.
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".