The Shulhan Aruch (Orah Haim 589) writes that the obligation of Shofar on Rosh Hashanah falls under the category of "Misvot Aseh She’ha’zman Gerama" – affirmative commands that apply only at specific times – and, as such, women are exempt from this obligation. Strictly speaking, then, women are not required to hear the Shofar on Rosh Hashanah. However, as noted by the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), women have taken it upon themselves to hear the sounding of the Shofar, even though they are not strictly obligated to do so. He adds that it was customary in his community in Baghdad that when a woman was unable to attend the synagogue on Rosh Hashanah, somebody would go to her home to sound the Shofar for her, so she could fulfill the Misva.
This custom mentioned by the Ben Ish Hai is noteworthy, for two reasons.
First, it shows that he followed the ruling of the Shulhan Aruch that one may sound the Shofar on Rosh Hashanah for women. The Peri Hadash (Rav Hizkiya Da Silva, 1659-1698) disagrees with this position, and maintains that it is forbidden to blow the Shofar only for a woman or group of women. Sounding a Shofar is forbidden on Yom Tob, and is permitted on Rosh Hashanah only for the purpose of fulfilling a Misva. The Peri Hadash thus contends that since women are exempt from the Misva, one who has already fulfilled the Misva may not sound the Shofar for a woman. The Ben Ish Hai clearly followed the Shulhan Aruch’s lenient position, that since women are, after all, credited with a Misva if they hear the Shofar, this suffices to permit blowing the Shofar for them on Rosh Hashanah.
Secondly, the Ben Ish Hai’s comments show that in his view, it is permissible to carry a Shofar through a public domain on Rosh Hashanah for the purpose of the blowing the Shofar for a woman. Tosafot (Talmudic commentaries by Medieval French and German scholars), in Masechet Hullin, maintained that carrying in a public domain is permitted on Yom Tob only for a "Sorech" ("need"), and blowing the Shofar for a woman does not qualify as a "Sorech" because they are not obligated to hear the Shofar. The Tur (Rabbenu Yaakob Ben Asher, 1269-1343) cites the Ba’al Ha’ittur (Rav Yishak Ben Abba Mari, 1122-1193) as disagreeing, and claiming that since a woman fulfills a Misva if she hears a Shofar, this qualifies as a "need" for which carrying is allowed on Yom Tob. In truth, all this is immaterial in light of the Shulhan Aruch’s ruling (Orah Haim 518), following the position of the Rambam (Rav Moshe Maimonides, Spain-Egypt, 1135-1204), that carrying on Yom Tob is allowed under any circumstances. According to Sephardic practice, then, there is no question at all whether a Shofar may be carried through a public domain on Rosh Hashanah for the sake of blowing it for a woman, since carrying is entirely permissible on Yom Tob (as long as the object is not Mukseh).
There is a famous debate among the Rishonim whether a woman who performs a Misva from which she is exempt recites a Beracha over the Misva. Tosafot, Rabbenu Tam (France, 1100-1171), and several other Rishonim maintained that a woman does, in fact, recite a Beracha, since she does, after all, receive reward for performing the Misva. And although the text of the Beracha includes the word "Ve’sivanu" ("and commanded us"), implying that even the woman is included in the command, these Rishonim explain that this refers generically to the command given to the Jewish People, and does not imply that the women are obligated. The Rambam, however, writes that if a woman wears Sisit, she does not recite a Beracha, because it is a "Misvat Aseh She’ha’zman Gerama" from which she is exempt – clearly indicating that a woman who performs a Misva from which she is exempt does not recite a Beracha. The Shulhan Aruch follows the Rambam’s position, and thus rules that a woman does not recite a Beracha over the sounding of the Shofar, since she is not obligated in this Misva.
Interestingly, the Hid"a (Rav Haim Yosef David Azulai, 1724-1806) concludes that despite the Shulhan Aruch’s ruling, women should, in fact, recite a Beracha over Misvot which they voluntarily perform. He notes a responsum by Rav Yaakob of Marvege (France, d. 1243), a Rabbi who would pose Halachic questions before going to sleep, and then receive an answer in a dream during the night. He compiled these responses in a work entitled "Min Ha’shamayim." One of the questions he asked was whether women recite a Beracha when performing a "Misvat Aseh She’ha’zman Gerama," and the response he received was that women indeed do recite a Beracha over such a Misva. The Hid"a asserts that if Maran (author of the Shulhan Aruch) would have been aware of this response, which was shown to Rav Yaakob of Marvege in a quasi-prophetic dream, he would have accepted this ruling. Accordingly, the Hid"a ruled that women generally do recite a Beracha when they perform a "Misvat Aseh She’ha’zman Gerama." He makes an exception, however, with regard to the Misva of Shofar, noting that even in Sephardic communities in which women recited a Beracha over other Misvot, like Lulab, women would not recite a Beracha over Shofar. This is also the ruling of Hacham Bension Abba Shaul (Israel, 1924-1998). Hacham Ovadia Yosef, however, challenges the Hid"a’s claim, arguing that we follow the Shulhan Aruch’s rulings even if they run in opposition to the positions revealed to Rav Yaakob of Marvege.
In any event, it is clear that according to all opinions, if one blows the Shofar only for a woman or group of women, no Beracha is recited.
The Ben Ish Hai writes that if a woman normally hears the Shofar on Rosh Hashanah – which is, of course, the commonly accepted practice – and one year she knows she will be unable to, then before Rosh Hashanah she must perform Hatarat Nedarim, annulling her implicit vow to observe this custom.
The Sha’areh Teshuba comments that it is preferable for a woman to hear the Shofar sounded by a man, rather than blow it herself or hear a woman blow it.
Finally, the custom that women ensure to hear the Shofar is limited to the first 30 Shofar sounds, which are blown in the synagogue before Musaf. A woman is not required – even by force of accepted custom – to hear all 101 sounds that are blown in the synagogue; it suffices to hear just 30 sounds.
Summary: Although women are exempt from the obligation of Shofar, it is customary for women to hear the first 30 sounds of the Shofar. If a woman cannot attend the synagogue, it is permissible, and proper, for a man to go to her home and blow the Shofar on her behalf. He may carry the Shofar through the public domain for this purpose. No Beracha is recited if the Shofar is being blown only for a woman or group of women. A woman who knows before Rosh Hashanah that she will be unable to hear the Shofar should perform Hatarat Nedarim.