The blowing of the Shofar is the central Mitzvah of Rosh Hashana and whoever is blowing it must do so standing up (Shulhan Aruch, O.H. 585:1). The Ashkenazim follow the custom that even the congregants listening to the Shofar should stand up (Magen Avraham 585:1). The HIDA (Birke Yosef) has difficulty with this practice since the Shofar is referred to as “Tekiot Bimyushav” (“seated blasts”) because one achieves better concentration in this position. Some rabbis take a middle approach and say that the congregation should stand for the blessings made over the Shofar but then sit to hear the Shofar blasts. Rabbi Baruch Toledano (Kitzur Shulhan Aruch) confirms that our custom is for the congregation to remain seated for both the blessings and for the blasts.
Summary: One should sit when listening to the Shofar, including its blessings.