The clearest source for the tradition of a tent with openings on four sides is Kad HaKemach, Guests 1:1, which cites a midrash stating that Avraham's house had four doors facing the four directions of the world, so that whoever entered through one door could leave through another, thereby sparing passersby embarrassment, and so that Avraham could receive travelers from every side.
Bereshit Rabbah 48:9 records Rabbi Abahu's statement that Avraham's tent was "mefulash" — open through-and-through — bringing Rabbi Yudan's parable of a man who, upon seeing travelers veer toward his house, knows they intend to visit him, which the midrash uses to explain how Avraham could be both sitting at the tent's entrance and running out to greet the three men, since the tent had passage from more than one side.
Etz Yosef on Bereishit Rabbah 48:9:12 sharpens this reading by explaining that the midrash's language of the tent being "mefulash" resolves an apparent contradiction between Avraham sitting at the entrance and then running toward the visitors: the travelers initially passed near one opening of the tent, and as they moved further along toward a second opening, Avraham turned from the first entrance and ran to meet them from the second, implying two functioning doors rather than a single entrance.
Midrash HaGadol and Zohar, Bereshit 12:140 both describe Avraham sitting at the entrance of the tent to receive guests, with the Midrash HaGadol explaining that this was not his ordinary practice but a deliberate act after two days passed without visitors, done so that passersby would not think he was displeased with them, while the Zohar describes the entrance itself as oriented "from below to above" — neither source, however, speaks of four doors specifically, so the four-sided tent tradition itself rests on Kad HaKemach, Guests 1:1 rather than on these two.