COVID-19 Vaccine Distribution and Production Act This bill requires the President to, within one day, exercise certain authorities under the Defense Production Act of 1950 to obtain supplies necessary to distribute and administer COVID-19 (i.e., coronavirus disease 2019) vaccines. That act confers on the President a broad set of authorities to influence domestic industry in order to provide essential materials and goods for the national defense. The President must allocate these supplies based on requests from governors and the needs of the states as determined by the population of each state. The bill's directive to use these authorities terminates when all requests from the states have been fulfilled. Supplies that have been produced but are unused at that time must be deposited into the Strategic National Stockpile.