The Shulchan Aruch (47:14) paskens that women make birchas hatorah. The obvious question is why, since they are exempt from the mitzvah of studying torah as we see from our gemara? The Magen Avrohom quotes the Agur who says that they are chayev in parshas karbonos and in the mitzvos that apply to them. Simply speaking the Agur means to say that although they have no mitzvah to study torah at all, they do have a requirement to know the mitzvos that are applicable to them. You can't know without learning and since they are forced to learn, they can make a bracha on that learning. The difficulty with this approach is that it doesn't justify how they can make "אשר קדשנו במצותיו וצוונו לעסוק בדברי תורה" if the bottom line is that they are not commanded to learn? The Beis Halevi (Teshuvos 1:6) raises this question (his approach doesn't seem correct - עיקר חסר מן הספר).
Another possibility is that the Agur is not simply saying that women have to know, which requires a them to learn so that they know. Perhaps he is suggesting that they actually have a mitzvah to learn the areas of halacha that pertain to them. According to the first approach if a woman is always around someone who can instruct her what to do there would be no purpose in her learning, but according to the second approach she has a mitzvah not only to know what to do when the situation arises but she has a mitzvah to learn all areas of halacha that pertain to her so that she will be proficient in them. Even according to the second approach, the value is not in the learning, rather in the knowledge to perform, but the nature of the mitzvah is to be proficient so that she will be able to perform the mitzvos better. Being that the nature of the mitzvah is to learn so that she will be proficient, it is understandable why she would be able to make birchas hatorah.
The Gr"a (and beis halevi) say that the only justification for a women to make birchas hatorah is to rely on Rabbeinu Tam (Tosafos R"h 34a) that women are entitled to make a bracha on mitzvos that they are not commanded in such as lulav, succah and tzitzis. Similarly, they can make bracha on Torah. This approach only justifies the bracha of אשר קדשנו במצותיו וצוונו לעסוק בדברי תורה. The Gr"a clearly holds that the bracha of אשר בחר בנו מכל העמים ונתן לנו את תורתו was never a question - that is a bracha of praise that can surely be made by women. The only point of discussion is how they can make the first bracha, and for that they need to rely on Rabbeinu Tam. This is clearly the approach of the Minchas Chinuch (end of mitzvah 430), that the bracha of אשר בחר בנו is an independent mitzvas aseh to make a bracha prior to learning, and the bracha of אשר קדשנו במצותיו וצוונו לעסוק relies on Rabbeinu Tam.The difficulty with the approach of the Gr"a and Minchas Chinuch is that the Shulchan Aruch does not hold like Rabbeinu Tam, he paskens like the Rambam that women can only make a bracha on mitzvos that they are obligated in.
The Brisker Rav (hilchos brachos) has an approach where he explains that birchas hatorah is not a birchas hamitzvah, rather it is a din that תורה בעי ברכה, meaning that one most show importance to torah by making a bracha on it prior to learning. This would certainly explain how woman can make a bracha, but it would logically only apply to the bracha of אשר בחר בנו, which is the Gr"a seems to hold was never a question to begin with. It is true that the brisker rav himself holds that his answer will explain both brachos, it doesn't seem to work for the bracha of אשר קדשנו במצותיו וצוונו which is the nusach of a standard birchas hamitzvah.
I would like to suggest based on the shulchan aruch (47:6) that the bracha of והערב נא should be said with a vav because it is not independent, rather a conclusion of the first bracha. It is clear from the gemara in sotah that although a woman is obligated to learn torah, she is obligated to provide emotional support to her husband and sons so that they excel in torah. The bracha of:
אשר קדשנו במצותיו וצוונו לעסוק בדברי תורה והערב נא ה' אלקנו את דברי תורתך בפינו וכו' ונהיה אנחנו וצאצאינו וצאצאי עמך וכו
Perhaps the וצוונו is not going just on the לעסוק בדברי תורה but rather on the entire bracha so that the bracha is not speaking of a command to learn torah personally but rather you commanded klal yisroel to study Torah, and commanded them to raise children that study Torah - this a woman is certainly commanded in.