Why the military? There are dozens, if not hundreds, of airliners and airlines available. Aren’t there?
There are temperature requirements for this specific vaccine that would be better accommodated by military aircraft. As mentioned above, potentially you can just put the refrigerated trailer on the aircraft. For civilian aircraft you would need specialist temp controlled cargo pods, which are very expensive and there's just not enough of then out there to meet the demand globally. Or you could maybe use several tonnes of dry ice, which is risky. This is about logistics. The requirements of the Pfizer vaccine are better met by Western nations that can store it at -80c. The J&J vaccine will be fine at room temperature so that's better suited to less developed regions and civilian aircraft.