The core source is the Gemara's dispute over whether the mitzva of the aravah was instituted as a binding ordinance or merely as a practice carried on by the prophets, where it is stated that the willow branch was either an ordinance of the prophets or an ancient custom of the prophets, and the Gemara identifies Chaggai, Zechariah, and Malachi as the prophets who instituted it in the Temple as obligatory according to the opinion that it is a binding ordinance (Sukkah 44a:9).
Rashi (Sukkah 44a:9) explains this directly, identifying the ordinance of the prophets as the enactment of the latter prophets Chaggai, Zechariah, and Malachi, who were among those who established ordinances for Israel together with the Men of the Great Assembly.
The Mishnah describes the practical mechanics of the mitzva in Temple times, where branches of the aravah were gathered at a place called Motza and stood upright beside the altar with their tops bent over it, and the altar was circled seven times on that day (Mishnah Sukkah 4:5).
Later authorities codify the halachic upshot of this dispute: the Rambam rules that since the aravah is not explicit in the Torah, it is taken only on the seventh day as a remembrance of the Temple, without a brachah, because the matter is a custom of the prophets (Mishneh Torah, Shofar, Sukkah and Lulav 7:22), and the Arukh HaShulchan likewise attributes the practice to the latter prophets Chaggai, Zechariah, and Malachi together with the Men of the Great Assembly, explaining that because it is not explicit in the Torah it was fixed only for the seventh day as a remembrance of the Temple (Arukh HaShulchan, Orach Chaim 664:2).
Rav Eliezer Melamed summarizes this same line, writing that the prophets instituted taking aravos not only in the Temple but everywhere, and that after the Temple's destruction the practice of taking aravos on the seventh day as a remembrance of the Temple continued, with the seventh day chosen because it was there that the primary mitzva of the aravah occurred in the Temple with the sevenfold circling of the altar (Peninei Halakhah, Sukkot 6:2), and he adds that the amoraim disputed whether the prophets established the beating of the aravah as a binding enactment requiring a brachah or merely as a custom without one, with the practical ruling following the view that it is a custom and no brachah is made (Peninei Halakhah, Sukkot 6:3).