Well, the authority comes from the generals and The Emperor himself(the Darth Vader-esque figure). The soldiers and Heartless are the muscle. And the generals themselves, when needbe, are too the muscle.
They still are, but yeah. The way I think it would have worked was that The Emperor showed up at Disney Castle one random day, ten years ago, and singlehandedly took out the top authority figures of Light. And for ten years, established a dictatorship(or tyrannic rule, whichever you prefer), building his Empire to the point where, as stated, most worlds within immediate reach were consumed by darkness. And he declared martial law against anyone who even dared to go against him and his empire.
This is where I think the Fighting Hearts came into the picture. Maybe they started causing trouble for the empire a year ago. So we can at least have some groundwork to build our timeline of sorts.