Women are exempt from the obligation of Shofar on Rosh Hashanah, in accordance with the general rule exempting women from "Misvot Aseh She’ha’zman Gerama" – Misvot that apply only at a particular time. As the obligation of Shofar applies only on Rosh Hashanah, women are exempt from this Misva.
Nevertheless, if a woman comes to the synagogue and hears the sounding of the Shofar, she is credited with a Misva. Although she has no obligation to hear the Shofar, she is considered as having performed a Misva if she chooses to do so. And thus a man who has already fulfilled the Misva is allowed to blow the Shofar again for his wife or another woman. This is not considered a violation of Yom Tob by producing sounds, because the woman fulfills a Misva by hearing the Shofar, and thus it is permissible to blow the Shofar for her just as it is permissible to blow the Shofar for men. In such a case, however, no Beracha is recited over the Misva, neither by the woman nor by the man blowing the Shofar. Women who perform a Misva from which they are exempt – such as sitting in the Sukka or taking the Lulab – do not recite a Beracha over the Misva, because the Beracha is only for those who are commanded to perform the Misva (as indicated by the text of the Beracha – "Asher Kideshanu Ba’misvotav"). Therefore, when one blows the Shofar for a woman on Rosh Hashanah, no Beracha is recited.
One may carry a Shofar through a public domain on Rosh Hashanah for the purpose of blowing the Shofar for a woman. Carrying on Yom Tob through a public domain is permitted "Le’srech Ha’yom" – when this is needed for the Yom Tob – and carrying a Shofar so it can be blown for women is considered a need that allows carrying. Although women are exempt, they nevertheless fulfill a Misva by hearing the Shofar, as discussed, and therefore carrying the Shofar to a woman’s home so she can hear the blowing is allowed. By the same token, it is permissible to carry the Shofar on Rosh Hashanah in a public domain to blow it for a child who has reached the age of Hinuch (training in Misvot), as this, too, is considered a legitimate need of Yom Tob.
Summary: Women are exempt from the Misva of Shofar, but they fulfill a Misva if they hear it blown. A man may blow the Shofar for a woman on Rosh Hashanah, even if he had already fulfilled the Misva, and he may even carry the Shofar through a public domain on Yom Tob for this purpose. However, neither he nor the woman recites a Beracha over the Misva, since she is not obligated in the Misva.
