The first time the term "Yetzer...Ra" is used is in Genesis 8. It takes place after Noah brings sacrifices after the Great Flood.
וַיָּ֣רַח יְהֹוָה֘ אֶת־רֵ֣יחַ הַנִּיחֹ֒חַ֒ וַיֹּ֨אמֶר יְהֹוָ֜ה אֶל־לִבּ֗וֹ לֹ֣א אֹ֠סִ֠ף לְקַלֵּ֨ל ע֤וֹד אֶת־הָֽאֲדָמָה֙ בַּֽעֲב֣וּר הָֽאָדָ֔ם כִּ֠י יֵ֣צֶר לֵ֧ב הָֽאָדָ֛ם רַ֖ע מִנְּעֻרָ֑יו וְלֹֽא־אֹסִ֥ף ע֛וֹד לְהַכּ֥וֹת אֶת־כָּל־חַ֖י כַּֽאֲשֶׁ֥ר עָשִֽׂיתִי:
And the Lord smelled the pleasant aroma, and the Lord said to Himself, "I will no longer curse the earth because of man, for the imagination of man's heart is evil from his youth, and I will no longer smite all living things as I have done.
Even earlier, in Genesis 4, reference is made to the Yetzer Hara:
הֲל֤וֹא אִם־תֵּיטִיב֙ שְׂאֵ֔ת וְאִם֙ לֹ֣א תֵיטִ֔יב לַפֶּ֖תַח חַטָּ֣את רֹבֵ֑ץ וְאֵלֶ֨יךָ֙ תְּשׁ֣וּקָת֔וֹ וְאַתָּ֖ה תִּמְשָׁל־בּֽוֹ:
Is it not so that if you improve, it will be forgiven you? If you do not improve, however, at the entrance, sin is lying, and to you is its longing, but you can rule over it."
ואליך תשוקתו: של חטאת, הוא יצר הרע, תמיד שוקק ומתאוה להכשילך:
ואתה תמשול בו: אם תרצה תתגבר עליו:
"And to you is its longing" [The longing] of sin, which is the Yetzer Hara, which constantly longs and lusts to cause you to stumble.
"but you can rule over it" If you wish, you will overpower it.
[Source: Sifrei Ekev 45, Kiddushin 30b]
So we have the Talmud and early Medrashim also speaking about the Yetzer Hara. Indeed, both the Talmud and Medrashim are peppered with references to the Yetzer Hara (just try searching "יצר הרע" in any Talmudic search engine. You can find a few examples here and here.)
As far as the identification of the snake with the Yetzer Hara, I believe the source is the Zohar.
"רבי יצחק אמר: דא יצר הרע, רבי יהודה אמר: נחש ממש" (פרשת בראשית דף לה ע"ב).
Rabbi Yitzchak says: This is the Yetzer Hara. Rabbi Yehudah says: An actual snake."
One of the medieval commentators explain that previous to the First Sin, the Yetzer Hara (=snake) was external to the person. After the First Sin, the Yetzer Hara was internalized.