This question is essentially communal, and its ramifications for the community’s identity, cohesion, and avodat Hashem play a major part in any decision. Arriving at a balanced decision entails consultation with a local religious authority who is in position to judge what will be of most spiritual benefit to the specific community. In cases like these, communal norms can have greater halachic weight than written sources. Our response is intended as a preliminary discussion of the general issues.
Rema OC 585:4 refers to calling out to the shofar blower (being makri) as a custom. On a strict halachic level, anyone who knows what should be blown — and how — can fill the role of telling the shofar blower what to blow.
Rema and other halachic authorities, however, express a preference for the makri to be the shaliach tzibbur (Darchei Moshe and Magen Avraham ad loc), or the community’s rabbi (Mishna Berura ad loc) or the greatest Torah scholar available (Aruch Ha-shulchan).
Thus, it would not be forbidden for a woman to fill this role, especially if she is a great Torah scholar within a community. However, a woman would usually not be the go-to choice, also because there may be reservations about a woman playing this role in the middle of tefilla (as opposed to, say, at a later blowing primarily for women). Ultimately, as above, this is a communal decision that depends on the nature and character of the community.
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