There's no real start-date to when the various Latin vernaculars "finish" morphing into the dialects of Italy (as in the rest of the Mediterranean, for that matter). Rather, languages change organically, and most watersheds are mere conventions. While it is a difficult and probably contentious to pinpoint a precise point where Latin vernacular becomes a dialect of Italy, we can set some broad chronological markers. In a similar vein, I wrote this answer about a month ago on the Dialects of Italy which might interest you.

In the specific instance of Sicilian, the there is a sort of "boom" of early Sicilian-language writing (almost all in a poetic or otherwise artistic capacity since the language for "official" state or legal business was still Latin) during the reign of Frederick II (towards the end of the first half of the 13th century). I don't know what the current consensus is on the oldest documented text that has been transmitted to us from that era, but it will probably date no earlier than sometime around 1220, when Frederick left Sicily to be crowned Holy Roman Emperor and left the Sicilian Kingdom in the hands of a circle of administrators who would take an interest in the literary arts. An interest in literature seems to have been itself rather novel and new: French troubadours had found a home in Sicily since the Norman conquest (and these performers had quickly adapted their performances to the local vernacular) so what we see during Frederick's reign is the adaption and evolution of this largely oral style into a written form by the local literate class.

The issue with studying this earliest form of Sicilian is that very few original documents have been preserved and passed down to modern scholars. Most of what is known about Sicilian writers of the 13th century actually comes from nearly a century later, when Tuscan writers discussed (and when we're lucky, transcribed) Sicilian literary works. Indeed, the largest compendium of Sicilian literature was transcribed by a Tuscan identified only as "Dante's Friend" (who slipped in a sonnet by Dante in addition to a few of their own at the very end of the compendium) who, however, also translated the sonnets into Tuscan as they were transcribed. So we know these sonnets were written in Sicilian in Sicily (Dante's Friend handily identifies authors by name, who are generally all traceable to the court of Palermo) without actually having trace of most of the originals as written in Sicilian.

We also of course can't exclude that a Sicilian literary scene existed before Frederick's court took an interest in literature and poetry, or that in any case there were people writing in Sicilian whose works have not been transmuted down to us. We also cannot discount early informal annotations and hasty transcriptions written in the centuries before Frederick that would qualify as some early form of Sicilian (there is really no way to draw a line between very vernacular Medieval Latin and early Sicilian, and this goes for any dialect of Italy, or any Romance Language for that matter).

So that's the long and short of it. A recognized circle of Sicilian-language authors emerged at the court in Palermo in around 1220. However, because most of what has survived from the Sicilian literary school exists in the form of Tuscan transcriptions, we do not have a whole lot of documentary evidence.