There's no definitive moment when a switch was made. But already at the time of the Roman Empire there were people speaking various vernaculars, and latin itself morphed stylistically over the course of Late Antiquity and the Middle Ages. But while Latin remained the language of legal affairs and church business, by the ninth century at least we know that there was sufficient distinction for vernacular to be considered a separate language from latin, not just the "vulgar" form of the same language. It's probable this separation actually happened much earlier, but the oldest inscription in a language identifiable as Italian dates to the ninth century, so that's the only real date to pinpoint. The inscription itself, Non dicere ille secrita a bboce, is a graffitto reminding visitors to speak softly or not speak at all in a Roman crypt. So when a disgruntled clergyman found it necessary to remind visitors in the crypt to be respectfully quiet, he believed it more productive to deliver this instruction in the vernacular.

A second attestation of vernacular involves a tenth-century legal deposition involving a land dispute with the abbey of Monte Cassino. For some reason, the notary taking down the deposition writes down one of the testimonies in the local vernacular in a document otherwise written in Latin. This probably means that legal practitioners would commonly translate testimonies from the vernacular into Latin. And while following century there's no real instance of the vernacular ever replacing Latin in official documents it does seep in from time to time, especially when referring to the structure and regulation the unique professions and practices that were tied to the trade guilds, which were becoming increasingly influential in the administration of Italian cities. Thus the imprecise latin term magister is replaced with the recognizably Italian title maestro, along with other words that don't exist in Latin like that referring the custody of guild secrets, "guardatura," while the growing business of insurance would also demand new words, for example referring to contracts providing for the "salvamento" of goods. The vernacular would also seep into legal documents when referring to political offices that had not existed in roman times, as well as in reference weights, measures, and coins introduced by the autonomous Italian cities.

The "formal" use of Latin has a bit of resurgence in the twelfth century. The unifying power of the use of Latin, both for symbolic and practical purposes, had been a recurring theme in efforts to centralize authority in post-Roman Italy and would appear again again with Frederick Barbarossa's legal and military campaigns against the increasingly autonomous Italian cities in the twelfth century. This theme has a long history prior to Frederick: King Theodoric's post-roman Kingdom of Italy made a point to follow Roman legal protocol and use of proper Latin as one of many successful legitimizing tools. The Byzantine occupation that would unseat the Ostrogoth monarchy similarly used Latin as a means of official communication. The Lombard ruling class that would replace the Byzantines also adopted Latin as the vehicle for the transmission of laws, while the Carolingian and post-Carolingian monarchy (such as it was) coupled religious investiture with political investiture, all the while also reinforcing the primacy of Latin in both those fields. It is only when the post-Carolingian crisis in Italy was decoupled with the post-carolingian crisis in the rest of Europe (for the system was nothing if not impressively unstable) that the domestic dimension of power struggles allows for the functional benefit of vernacular begin to seep into legal latin.

Prior to Frederick, there had been a resurgence of latin after the Ottonian conquest of Italy and creation of the Holy Roman Empire uniting what had been the German and Italian parts of Charlemagne's empire. But the vernacular again seeps into "official" latin under both the tutelage of the italophile Ottonian rulers andas well as the Salians, their successors who largely left Italy to its own devices. But when Emperor Frederick II, called Barbarossa, set out to curb Italian autonomy in the second half of the twelfth century, Latin again became a vehicle of law and legitimacy. Barbarossa and the legal scholars who argued in favor of his rule in Italy awakened a clear interest in the technicalities of the Roman Law which was still purported to be the legal system in Italy, and this sparked a new need for the use of proper Latin in official records and documents. Legal scholars in places like Bologna and Pavia took an enormous interest in the minutia of Roman Law and its language, not in the least to contest old claims being put forth by the Emperor and eventually lay new legal arrangements for the status of the Italian cities within the Empire.

Indeed, one of the surprising consequence of Imperial governance over Italy is a sort of "Legal Renaissance" which sees a trend of renewed study and imitation of old Roman legal texts.

But inevitably, the strain of conducting official business in a dead language would be impossible to bear. Once free from Imperial domination, in the thirteenth century there is a steady and irreversible uptick in use of local vernaculars in documents. Unsurprisingly, the first to permanently adopt the vernacular are merchants and shopkeepers (given to practicality as a matter of necessity) who prefer the vernacular for their ledgers and inventories. This led tax assessors to become some of the first early adopters of vernacular within government, with the remaining officers of government soon to follow. While examinations to become a notary and lawyer would require near-fluent knowledge of latin, there was nothing to be done about a preference for the vernacular. You do see things like legal documents written in the vernacular referencing re-written "official" copies in Latin, however it is often the vernacular version that has survived to the modern day while the latin version has been lost (a demonstration of which version record keepers found more useful to keep track of).

In the fourteenth century you begin getting extremely famous literary figures like Dante and Petrarch writing elaborate linguistic treatise in favor of the use of local vernaculars over Latin. The process by which the vernacular had gained precedence over Latin had ended, however the process through which the Italian vernacular was standardized had just begun. You can read about that second process here