Does halakhah permit a woman to affix a mezuzah?
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The Sefer Yeshouas Malkoi (פ''ה מהלכות מזוזה הלכה י) says She may not put up a Mezuzah. He brings a proof from a Gemara(מנחות מ''ב) that says a gentile can not sew, or work out the leather of the Tefillin as he may not write a Mezuzah so to he may not do the other jobs, therfore a women because they too can not write can also not put up a Mezuzah.
The Chasam Sofer in his SH"UT(יור''ד סימן רע''א ד''ה ולדידי) disagrees and says she may put up a Mezuzah and that Gemara is only talking about the production aspect the things that are learnt out of you shall write them but not the putting up of the Mezuzah as it is a side matter.
The SH"UT Eretz Tzvi(חלק א' סימן ט''ו) brings another point it would not make sense that you can make a Shliach to do somthing you yourself can not do.Known as
The Halacha is as the Chasam Sofer and it says in SHU"T Shevet Halevi (חלק ב' סימן קנ''ח ד''ה ג' ודע) who says it is allowed based on the fact that it can not be she can make a Shliach for somthing and can not do it herself.
The Sefer Daas Kedoshim (רפט) in discussing the mitzvah of affixing the mezuzah has a doubt if it was put up by a minor.
The sefer Mikdash Me'at (שם ס"ק ב) brings the words of the Magen Avraham (ס' יט ס"ק א) which address clearly that the main aspect of the mitzvah of mezuzah is not the affixing, rather the mitzvah is to live in a home that has a mezuzah, therefore why does it matter of it was affixed by a child or a gentile? This is analogous to a Sukka that was built by a gentile, that despite the gentile being totally removed from the Mitzvah of Sukka, nonetheless it is Kosher.
The Maharam Shi"k (או"ח ס' טו) echoes this idea, that the mitzvah is for the doorway to have a mezuzah, even if it was affixed by someone who is not a "בר חיובא" - obligated in the mitzvah.
Now, all of the above is speaking about minor and a gentile. Both a gentile and a minor are exempt from the Mitzvah of mezuzah. However a woman is absolutely obligated in affixing a mezuzah on her doorway as the Rambam writes (ספר תורה מזוזה ח: י) succinctly:
Therefore she is better than both a gentile and a minor regarding her level of obligation. And following the logic above, that the mitzvah of mezuzah is not the affixing, but that the doorway should have a mezuzah, there would be no issue for a woman to affix the mezuzah.