In Plato's Republic, where can the winged chariot story be found?

Answer 1

The passage cited in Joseph's answer is not from the Republic, it is from a commentary on the Republic. The "winged chariot," driven by Zeus, is from the Phaedrus.

Answer 2

I found for you this:
And not only does the vocabulary used to describe the intrusion of Polos, and later that of Callicles, is consistant with the "part" thet play, but their very name is too! Polos means "young colt", and reminds us of the winged chariot used as an image of the soul in the Phædrus. And Callicles means "one who as a reputation for beauty", without it being possible to decide whether this reputation is deserved or not, the word "kleos" having both meanings. And don't get me wrong! I don't pretend to settle the question whether these people are real persons from Socrates' time or merely Plato's creations. Even if they were real, nothing prevents Plato for choosing them, among other reasons, for the relationship betwen the meaning of their name and the part they are to play in the dialogue (for instance, I strongly believe that the Aristotle, one of the thirty, chosen as respondant in the second part of the Parmenides, who is a historic person, was chosen by Plato because he had the same name as a more famous philosopher, Aristotle, the pupil of Plato, and Plato was discussing in this dialogue the precise point where he thought his pupil could not understand him). Back to the Gorgias, for sure, Gorgias is a real person; this is less sure for Polos, and not sure at all for Callicles. Yet, even if both were real people, there remains to decide why Plato chose precisely those persons to play a part in his dialogue. On Socrates' side, Kairephon means "the one who manifests hapiness, joy, through his speech", a joy of the whole person ("Kaire" was Plato's greeting in his letters). And this is appropriate for one who is only there to let his logos talk, and merely create the circumstances of the discussion, like a tamed thumos in a just soul in the manner of the Republic
More you will find here:
http://plato-dialogues.org/email/950212_1.htm

and here
http://www.wku.edu/~jan.garrett/pgods.htm

and here
http://www.wutsamada.com/alma/ancient/chariot.htm

Answer 3

The first reference to the subject you can find it in Fedro, after that Timeo and also in the Republic, but there you'll find mainly corrections.