Simlah Chadasha writes (2:25) that those groups of people that do not accept the validity of the Oral Torah such as צדוקים (Sadducees) is acceptable, because even though all laws of kosher slaughter are recorded in the Oral Law and no mention of them is made in the Written Law, and the Sadduccees rejected the Oral Law; nevertheless, since it is clear that the laws of kosher slaughter are contained outside of the Written Law*, the slaughter of a Sadducee is acceptable (if we have established that this particular person knows the laws -- see there) because they have the same tradition of slaughter laws that we do.
What is a little confusing is that at the end of that halacha, after writing that the shechita of Sadducees is acceptable, Simla Chadasha writes "therefore, the shechita of Karaites is not acceptable." [emphasis mine]
What follows is from my rebbi's notes on Simla Chadasha, answering this question:
The Karaites are different from the Sadducees and the Bitusim. While the latter are allowed to slaughter for a Jew, if they are being adequately supervised, the Karaites are not. The Karaites who lived at the time of the Simlah Chadashah, did not accept the oral laws, even those pertaining to shechitah. As a result they were not considered to be בר זביחה and that which they slaughtered is neveilah. Once a person is not a בר זביחה the meat this person slaughtered is forbidden regardless of the level of certification. The meat is neveilah even if every action of this person were observed to be correct and the knife was inspected prior to the shechitah and found to be perfect.
Question 1: Is the above quote accurate? How do Karaites slaughter animals to avoid the prohibitions of neveilah נבילה (Deut. 14:21) and treifah טריפה (Ex. 22:30)?
Question 2: How would a Karaite explain Deut. 12:21, "וזבחת...כאשר צויתיך" "and you shall slaughter...how I commanded you" -- where is that צווי if it was not written down anywhere in the Torah?
* Because it is written (Deut. 12:21) "וזבחת...כאשר צויתיך" "and you shall slaughter...how I commanded you." The problem is that there is no mention of a method of slaughter recorded anywhere in Tanach. Therefore, that this verse teaches that Hashem taught Moshe about the method of kosher slaughter on Mount Sinai. (Chullin 28a, SC 1:1)