What is the halachic source requiring us to ritually slaughter chicken specifically, and was there ever a time in history that Halachic Jews ate chicken without first performing ritual slaughter?
There is a documented opinion among Tannaim that there is no biblical requirement for Shechita on fowl (eg. Chullin 27b - 28a). Accordingly it would seem likely that some time existed before the rabbinic requirement of Shechita on fowl was enacted when Halachic Jews ate fowl without Shechita. I note that we do not rule like this opinion, but rather that Shechita on fowl is a biblical requirement.
As with all the biblical laws of Shechita, the source is oral tradition from God via Moses, as it says regarding Shechita in the Torah (Deut 12:21):
וזבחת... כאשר צויתיך
And you shall slaughter it... as I commanded you
and there is no written record of that command.