When they are running, the carriages, omnibus, and horse drawn trolleys stop and pick up passengers on the circles in front of the Main Street Station and the Castle. You don't hail them, you get in line.
The scenes you see in the commercials of the happy family riding the carriage up main street are filmed before the park opens. The vast majority of guests proceed up and down Main Street on shank's mare (on foot). That way you have the leisure to stroll and browse all the shops that line both sides of Main Street. The pedestrian traffic is usually so dense there really isn't much room for anything else.