I’m wondering what Halachah says about women blowing the Shofar? Is it permitted?

The Shulchan Aruch (OC 589, 6) writes that women can blow the Shofar for themselves, however should not recite the Brachah. The Rema ibid adds that our custom is they do say the Brachah, just like any other time-bound positive Mitzvah women are exempt form. The Sephardi custom is as the Shulchan Aruch writes and the Ashkenazi one – like the Rema.