A protagonist is the main actor of the story. He's the guy we root for, the one we want to see succeed in his mission, and the one we connect closest to. People often mistake this character role with the hero because, hey, heroes are often protagonist. In Foundation, the hero and the protagonist happen to be the same person, but they aren't the type of heroes to beat up the bad guys. Instead, they're the type of heroes who outthink them.
Salvor Hardin is the protagonist of both "The Encyclopedists" and "The Mayors." His ultimate goal is to prevent the Foundation from being conquered by the Four Kingdoms surrounding it. Most readers aren't exactly thrilled by the idea of war and death for the sake of conquest, so it's a safe bet they'll root for Hardin. Hardin succeeds at his goals by pursuing the path of nonviolence and rational thinking. Always commendable, that.
Hober Mallow is set up to take the fall as a traitor. Since he's anything but a traitor, we want to see him succeed at clearing his name. Mallow then becomes mayor and faces the third Seldon Crisis. In keeping with Foundation's themes, Mallow manages to prevent the war through nonviolent means. He's the guy who puts the plans in motion to prevent the war—i.e., the chief actor—so go ahead and chalk him up as a protagonist.
Yeah. Limmar Ponyets doesn't exactly get the biggest story in Foundation, but he's still a protagonist. His task is to free a friend from certain death. How can you not get behind that motivation? Plus, Poynets's ability to outmaneuver his opponents really gets you excited to see him come out on top.