While it's been the tradition for the Chief Justice to swear in the President, it's not required by the Constitution. In fact there's no requirement that anyone swear in the President. But he does need to swear the oath (Art II, Section 1, last paragraph) , as do other federal, state and local officers and officials (Art. VI, 3rd paragraph) He should already have sworn the very similar oath when he became a Senator, but I've read that many Congress Critters neglect to sign the documentation. (You try that and you'll never be a 2LT or Ensign, or even an E1)
If it came to that, I'm sure Ruth Buzzi Ginsberg would do the job, if she could stay awake long enough. Then she'd resign.
You know, any congressman or senator should have to take the oath, and the President. If the military does, they certainly should have to or --no job!