Quite often there was no Vice President. According to Cardinal Matthias the role of Vice President included responsibility for dealing with offworld representatives and security.
Matthias tried to trick both Romana II and Valyes into granting him the powers of the vice presidency during the emergency following the Gallifreyan Civil War. Because there had not been a Vice President for a long time, both Valyes and Braxiatel assumed the presidency through being Chancellor. (AUDIO: Appropriation, Mindbomb)
Saran was promoted to the position of Lord President from Vice President after the High Council voted unanimously to depose Morbius based on the findings of the Committee of Enquiry. (PROSE: Warmonger) Castellan Kelner was told to consider himself Acting Vice President by the Fourth Doctor during the Sontaran invasion of Gallifrey, though he continued to repeatedly address him as Castellan thereafter. (TV: The Invasion of Time)