Probably not. That would ruin the name of the Uchiha in the sense that, if it was allowed by nature that a woman can awaken the sharingan, what if the woman marries a man without the Uchiha name? She'll get the man's family name right? Then that would make a person without a family name of Uchiha, have the Sharingan.
For example, an Uchiha woman named, Lady Uchiha, has the sharingan. She later marries a man from the Sarutobi clan. So she'll be named Lady Sarutobi, right? Then it would appear that a person without the surname of the Uchiha, has the sharingan.
So let's suppose the examples had a child and inherited the Sharingan from the mother. So now that would appear that a child from the Sarutobi inherited the Sharingan.
So I theorize that there is no woman that has an awaken sharingan. Uchiha women cannot inherit/awaken the sharingan and also, Uchiha women cannot pass on the sharingan.The latter seems more plausible.
It's not sexism. Kishi isn't a chauvinist. It is only logical if he made women deprived of the ability to awaken/inherit/pass on the Sharingan.
Uchiha woman may have the ability to awaken or inherit the Sharingan, but Uchiha woman passing on the Sharingan is highly unlikely.