Can a female slaughter animals according to Halacha? (sources)
First siman in Yoreh Deah 1:1- They are kasher to shecht. The Rema quotes the Bais Yosef/Agur that we don't allow them lechatechila since we see the minhag is not to shecht.
(There is discussion on why the fact that women don't shecht creates a minhag for them not to.)
The first Tosafot in chulin entertains the idea. The Rosh permits it. Eldad HaDani, a big forger basically prohibits it, but some Rishonim thought he was for real. More information can be found in this class by Rabbi Maybruch. Rav Soloveitchik said that the only reason that we never saw women shochtot is because it used to be a position of communal responsibility (one was the shochet of a particular town), but now that it is not such a position, women can in fact do shechita.
The book Simla Chadasha (a popular Ashkenazi halacha book on shechita) says (1:13) that while men and women have the same status according to Halacha, the custom in his area is to not let women shecht even if they are experts in the laws and one should not give them kabbala (formal permission) to shecht. In such a place, he says, their shechita would only be kosher if a trained shochet watched her the whole time and checked her knife, or if he tested her afterwards in the laws of shechita and heard her orally testify that she did not become faint during the entire process.
In practice, women do not perform kosher slaughter.
Here are some highlights from the linked discussion:
"...the Agur says that the Minhag is that women do not Shect. The Bais Yosef asks, how can there be a Minhag that they don't Shecht. Maybe no women wanted to Shecht? Only if a woman wanted to Shecht and we wouldn't allow her would be a proof that there is a Minhag not to Shect. The Shach answers that the Agur holds like the Mahrik that by Minhagim we say that if we don't see something done we can still establish it to be a Minhag."
"The Simla Chadasha says that if a woman Shechted, we must assume that it's not a good Shcita until proven otherwise. Since she's not doing like the Minhag, we cannot apply to her the rule that most people that Shecht know how to Shecht, since she is not like most people, since she's not obliged to the tradition. For this reason, even if an institution gives her Kabalah, since they're going against tradition, they are also suspect."
Consider that, for centuries, people raised their own poultry in their backyards. Both Jews and Gentiles conformed much more strictly to normative gender roles back then; so we can assume that the wife / mother of the family was in charge of domestic needs, including cooking.
The woman would pluck the chicken carcass, clean it, spice it, and cook it. Yet, there was never a single Jewish community in which the same woman who dealt with all the blood and guts of "dinner", would actually shecht it herself. That sets a very strong precedent for Minhag Yisrael.