Disney only uses there own buses for the fixed destinations within the grounds. Those run all day, every day and are staffed with the needed numbers of drivers. They would cover the resorts, theme parks, water parks and DtD.
Anything that is just a once a day thing like the Backstage Tour, or longer distance runs like cruises, Magic Express, or, perhaps, this one you are referring too, would have Mears contracts to cover them. The buses, even those that have some Disney paint schemes, (Cruise or ME) are buses that are owned by Mears and leased, with driver, to Disney.