 Procedurally, Biden would either need to resign, but this would make Kamala Harris the president for the rest of the administration's tenure and likely make her the candidate, not Michelle Obama. Or he would need to declare that he is ending his campaign and release his delegates acquired through the primary process, where it would be the job of the delegates to throw their support to a new candidate, one of whom could be Michelle Obama. In the latter scenario, Michelle Obama would have to compete against other candidates.