No, you have to use "whom" in this sentence. But you can also remove it altogether, that's usually what happens in oral English. By the way, you have to put an "a" at the end of "know" and "door" takes 2 "o" :) So, possible correct sentences would be: *A woman whom my sister knows has bought the house next door. *A woman my sister knows has bought the house next door. (more natural)

In spoken English, you would almost always hear 'who'. The only exception would be during a formal speech or a lecture given by an English professor. It is wrong grammatically, but that almost never stopped anyone from using it - including me.

A woman (that/whom) my sister knows has bought the house next door. You can say "that" or "whom". I'm afraid "who" is not correct.