Growing up, I typically went to a Chabad and I have noticed that they have hakafot on erev Shemini Atzeret (meaning the night of Shemini Atzeret). What is the reason and why do they do this?
The Baal HaTanya writes in his siddur:
מנהג ותיקין הוא לעשות יום שמיני עצרת גם כן כמו בשמחת תורה
It is a custom of the ancients(?) to do on Shmini Atzeres like on Simchas on Torah ...
And then going on to describe Hakafos.
Rabbi Nachum Greenwald notes that the language seems to paraphrase the Mishnas Chassidim, but the Mishnas Chassidim is talking about Shimini Atzeres in Eretz Yisroel.
He then goes on to note that the custom of doing it on both nights is common among all Chassidim (not specifically Chabad) as this was the custom of the Mizritcher Maggid. In terms of source, he also points to the Tur Borekes (by a student of Rabbi Chaim Vital) who argues specifically for Hakafos at night having a special connection to Shimini Atzeres specifically (more so than Simchas Torah night) kabbalistically, but feels inadequate to actually introduce the practice.
There is a minhag to do it then to feel more connected with the Jews in eretz yisrael.
Source : siddurim and Chabad shul.