There was no written constitution. The system relied on written laws, unwritten rules and precedences that were handed down over generations, tweaked and changed over time, often in response to conflicts between the rich and poor citizens. The sources we have for the early republic are not great (we pretty much only have Livy, who wrote centuries later), so knowing exactly how and when the various institutions developed is very difficult. By the 2nd century BC, when Polybius was writing, the system was fairly stable, and his account in book 6 is the fullest description of how the system worked in the mid-republic. There were laws and codes of conduct and practice written down - especially decrees of the senate - and on one of the earliest bits of writing we have from Rome is an inscription that might be defining some of the roles of the early kings (the Lapis Niger). Even these law codes were not collected together in any form we would recognise today until much later in the Imperial period.