In any set of religious observances there are practices that are scriptural - as found in the bible, there are practices mandated by revered sages - the first few generations of rabbis who wrote commentaries on the Torah and oral tradition as elucidated in the Mishna, and there are practices based on tradition that grew out of the dim fog of historical past - such is the ritual of kaporos. A complete detailed discussion of the practice can be found here.
The prayer recited during the ritual is a combination of verses from the Hebrew Bible (Tanakh);
(After the ritual is completed the chickens are then taken to a kosher butcher for preparation and then donated to charity as food for the needy):
Children of man who sit in darkness and the shadow of death, bound in mystery and chains of iron --- He will bring them out of darkness and the shadow of death, and will sunder their bonds. Foolish sinners, afflicted because of their sinful ways and their wrongdoings; their soul loathes all food and they reach the gates of death --- they cry out to the Lord in their distress; He saves them from their afflictions. He sends forth His word and heals them; He delivers them from their graves. Let them thank the Lord for His kindness, and [proclaim] His wonders to the children of man. If there be for a man [even] one interceding angel out of a thousand [accusers], to spend of his uprightness in his behalf, then He will be gracious to him and say: Redeem him from going down to the grave; I have found expiation [for him].
This is my exchange, this is my substitute, this is my expiation. This chicken shall go to death and I shall proceed to a good, long life and peace.