“The one who blows the shofar on Rosh ha-Shanah…should likewise be learned in the Torah and shall be God-fearing, the best man available. However, every Jew is eligible for any sacred office, pro¬viding he is acceptable to the congregation. If, however, he sees that his choice will cause dissension, he should withdraw his candidacy, even if the improper person will be chosen”
(Shulhan Arukh 3:72). The Ba’al T’kiyah shall abstain from anything that may cause ritual contamination for three days before Rosh ha-Shanah (Shulhan Arukh 3:73).

When I train Shofar Sounders, I make sure that they all participate in the service and that no one is highlighted at the expense of another. I have also declined invitations to sound the shofar at other congregations if I know that I will displace another sounder. The democracy of each congregation is impor¬tant. And it only enhances a sense of participation of as many people as possible can participate in a service.
Finally, the Ba’al T’kiyah shall recite the benedictions before the shofar ceremony.

Arthur L. Finkle, Easy Guide to Shofar Sounding, Torah Aura, 2002

Moreover, the Talmud indicates that a woman can sound the Shofar but only for other women because this positive mitzvah (obligation) is time bound for which women are exempt.. Mishnah Berurah 588:6.

To which I would now add that, in the preponderance of non-Orthodox synagogues, women have achieved ritual equality as men. Accordingly, there is no prohibition for a man to sound the Shofar in these synagogues.

Arthur L. Finkle

The general rule is that women are exempt from mitzvahs which are time-dependant.1 This would include the Mitzvah of hearing the Shofar — which has a very specific time designation; it must be fulfilled during the daytime hours of Rosh Hashanah.
Nevertheless, there are certain time-contingent mitzvahs which women have accepted upon themselves to observe. Hearing the sound of the shofar is one of these. Indeed, there are Halachic authorities which maintain that because it has become a universally accepted custom for women to hear the shofar, today it is has become mandatory — as is the case with any custom which has become accepted practice.