The Spanish Inquisition were a hardline religious sect, intent on purging Heretics from the Spanish people. In game it functions as a separate faction, competing with The Knights Templar. Its agents and missions can be found in Santiago, San Juan, and Santo Domingo. Sinking the Dread Saints ships also increases the reputation.
Here is the list of missions that include Inquisition reputation as a reward. Pictured is Brother Mandez, The Inquisition exchange in Santiago.