Palpatine had a successor: Vader. And later he tried to recruit Luke Skywalker.

Since the modern Sith master-apprentice paradigm became established, the generally accepted course of events is that the apprentice eventually usurps the master and takes on an apprentice of his own. This even may have happened, if Luke had accepted Vader's offer on the catwalk of Cloud City.

However, all of that is moot, because Palpatine had previously been the apprentice of Darth Plageius, who had discovered Dark Side techniques for the manipulation and extension of life. Palpatine very likely knew a method of extending his natural lifespan to the point of cheating death. It's only by the intervention of Luke managing to turn Vader that he was killed through violence.

A Sith always had an apprentice. Once the apprentice felt they had mastered their power, they would battle their master to the death. In this case, Vader was setup as Palpatines successor. As Vader did in fact overthrow Palpatine (quite literally in this case) he did seize the thrown, however, his injuries in that battle ultimately took his life before formalizing his reign. This lead to a power vaccum that led to decades of imperial infighting with legend suggestion challengers such as Admiral Thrawn attempting to take the mantile upon himself. There is currently debate as to whether Thrawn was even a real person however, or simply a made up antagonist told to scare the children of the New Republic. There is a new documentary coming out this Christmas that claims to shed some new light on the post Palpatine Empire, though I am skeptical anything of value will be learned