Federal law mandates that a commercial driver's license or CDL, issued by a state (or territory or the District of Columbia) is required to operate
Class A: Any combination of vehicles which has a gross combination weight rating or gross combination weight of 11,794 kilograms or more (26,001 pounds or more) whichever is greater, inclusive of a towed unit(s) with a gross vehicle weight rating or gross vehicle weight of more than 4,536 kilograms (10,000 pounds) whichever is greater.
Class B: Any single vehicle which has a gross vehicle weight rating or gross vehicle weight of 11,794 or more kilograms (26,001 pounds or more), or any such vehicle towing a vehicle with a gross vehicle weight rating or gross vehicle weight that does not exceed 4,536 kilograms (10,000 pounds).
Class C: Any single vehicle, or combination of vehicles, that does not meet the definition of Class A or Class B, but is either designed to transport 16 or more passengers, including the driver, or is transporting material that has been designated as hazardous under 49 U.S.C. 5103 and is required to be placarded under subpart F of 49 CFR Part 172 or is transporting any quantity of a material listed as a select agent or toxin in 42 CFR Part 73.
The specific classification of license you need is dependent on the state, but you are required to have a CDL with sufficient endorsements if the school bus is designed to seat 16 or more passengers, even if there are a fewer than 16 of you.
If your group is indeed that large, you can rent large vans or mini-buses configured to seat up to 15 (e.g. Ford Transit, Mercedes-Benz Sprinter). Just search for "15-passenger buses" or shuttles, designed specifically to fall under the limit. If you have your heart set on the iconic yellow school bus, a quick web search did turn up a few companies that have configured school buses to seat 14 or 15, though these seem to be aimed at in-town excursions (e.g. youth groups, bachelor parties), and I cannot say whether any would be willing to rent you one for a cross-country trip.