No, the Torah does not mention explicitly that the Jews built actual Succot. (If it had, there wouldn’t be an argument as to whether the Jews had real Sukkot or Sukkot made from the Ananei Hakavod). However, in Vayikra 23:43, HaShem mentions the fact that the Jews lived in Succot, as the reason to celebrate Sukkot:
לְמַעַן֮ יֵדְע֣וּ דֹרֹֽתֵיכֶם֒ כִּ֣י בַסֻּכּ֗וֹת הוֹשַׁ֙בְתִּי֙ אֶת־בְּנֵ֣י יִשְׂרָאֵ֔ל בְּהוֹצִיאִ֥י אוֹתָ֖ם מֵאֶ֣רֶץ מִצְרָ֑יִם אֲנִ֖י יְהוָ֥ה אֱלֹהֵיכֶֽם׃
in order that future generations may know that I made the Israelite people live in booths when I brought them out of the land of Egypt, I the LORD your God.
In fact, the Gemara that is quoted brings this passuk as the backing point of both Rabbi Eliezer and Rabbi Akiva (Sukkah 11b):
הניחא למ"ד ענני כבוד היו אלא למ"ד סוכות ממש עשו להם מאי איכא למימר דתניא (ויקרא כג, מג) כי בסוכות הושבתי את בני ישראל ענני כבוד היו דברי ר' אליעזר ר"ע אומר סוכות ממש עשו להם הניחא לר"א אלא לר"ע מאי איכא למימר
The Gemara asks: This works out well according to the one who said that the sukkot mentioned in the verse: “I made the children of Israel to reside in sukkot” (Leviticus 23:43), were clouds of glory, as it is reasonable that the roofing of the sukka is modeled after clouds. However, according to the one who said that the children of Israel established for themselves actual sukkot in the desert, and the sukkot of today commemorate those, what can be said? According to that opinion, there is no connection between a sukka and a cloud. As it is taught in a baraita that the verse states: “I made the children of Israel to reside in sukkot”; these booths were clouds of glory, this is the statement of Rabbi Eliezer. Rabbi Akiva says: They established for themselves actual sukkot. This works out well according to Rabbi Eliezer; however, according to Rabbi Akiva what can be said?
(Quotes and translations from sefaria.)