It is reasonable to assume that they would, but this is not always the case. I guess that the availability of a shuttle is determined by (1) Marriott's policy regarding it's owned or operated hotels, (2) the franchise agreement or (3) what the local market dictates.
Also, as others have suggested, it is always prudent to confirm with a hotel that offers a shuttle, the exact operating hours and whether the shuttle operates on demand or simply periodically circles the airport's arrival areas.