Luach Dvar B'Itoh
According to the Luach Dvar B'Itoh it's a Chassidish Minhag to blow after each round of the 7 Hakafot on Hoshana Rabba.
His sources it from the Shloh HaKadosh, and other sources that I can't decipher unambiguously, if at all.
חס' לתקוע תשר"ת (ולמ"צ: תשר"ת תש"ת תר"ת. מ 365 ) בין "הושענא"
ל"הושענא" (ומקורו בסדה"י ובשלה"ק. שו"ת עטי"ש בסוף אמ"נ מועדים, ב)
His source is מ 365 which is:
הרב שלום שטמר, עורך "צאנז" הי"ו (נתניה) (as per page 1475) - the editor of the Tzanzer weekly.
It's possibly a reminder of the Shofar blowing that took place in the Bet haMikdash when they brought in the new Aravot daily and placed them around the Mizbeach, as documented in the Mishna in Suka 4:5
Wikipedia claims: " Among Sephardi Jews, .... In Amsterdam and in a few places in England, America, and elsewhere, the shofar is also sounded in connection with the processions. The latter practice reflects the idea that Hoshana Rabbah is the end of the High Holy Day season, when the world is judged for the coming year."
However, I did not find a mention of it in the קיצור ש''ע ילקוט יוסף, so they may be confusing Sephardi with Chasidish/Nusach Sephard.