This is a complicated question. If you are asking about the formation of a Republic in the sense of the Greek or Roman systems, whereby power rested with citizens/a senatorial class who made decisions and appointments through a voting process - I am not aware of any having existed in the Medieval Period.

There were, however, many different kinds of Islamic Government in the Medieval Period, which I will try to give a brief highlight of below.

Islamic political philosophy changed significantly over the course of the Medieval period - it began with personal control by Mohamed and his family, but as the religion spread across North Africa, West Asia and Europe different institutions began to emerge. The Golden Horde, for instance, reflected it's Mongol background by having a semi-elected Monarchy, whereby the Khan would have to be chosen by a kind of consensus process and then later evolved into being a purely hereditary monarchy. The same process happened in some caliphates - originally the Caliph was chosen through significant consultation with members of the ruling class, but eventually the position became hereditary. In the Fatimid State, which spanned all of North Africa and much of the Levant during the 10th century, most government positions were not hereditary and instead were merit-based, though the position of Caliph was hereditary. The Fatimid state also had some other things we traditionally associate with Republics, such as widespread freedom of thought, and a focus on individual rights/liberties. Having access to Greek texts through Egypt, and with great openness for new ideas, the Fatimid state may have been one of the most intellectually vibrant cultures of the entire Medieval Period.

One important idea in Muslim political thought throughout the Medieval period is "shura" - usually translated as "consultation." The basic idea is that it is beholden on anyone making a decision to consult with people who may be affected by this decision. This has been variously interpreted as consulting widely with advisers, other leaders, or members of the public depending on the situation/time period.

Sources:

Riley-Smith, Jonathan. The Crusades: A History. Yale University Press, 2005. Malouf, Amin. The Crusades Through Arab Eyes. Al Saqi Books,1985. Hourani, Albert. A History of the Arab Peoples. Grand Central Publishing, 1991.