The Mishnah in Rosh Hashanah says that on Sukkos is the judgement for water.
On four points the word is judged...and on Chag (Sukkos) we are judged on water.
The Talmud there explains that the special water libation that was brought on Sukkos was brought so that we would be blessed with rain in the coming rainy season (which occurs after Sukkos).
Rabi Yehudah said in the name of Rabi Akiva "...and why did the Torah
say to pour the water on Chag (Sukkos), the Honly One blessed be He said,
'pour before Me water on Chag so that you the rain of the year is
blessed for you.'
This relates to the reason we bring the four species as is explained in the Talmud in Taanis:
Rabi Eliezer said, 'Since the four species are only to appease on the
water. Just like these four species can't be had without water, so to
the world can't exist without water.'
As we can see, one of the major themes of Sukkos brought in the Talmud, is rain. Sukkos comes right before the rainy season. On Shemini Atzeres as well, we start mentioning rain in the second blessing of the Shemoneh Esreh (according to another Talmudic opinion, we should be starting on Sukkos). We have the special water libation as well. So taking the four species is very appropriate for the time.