The simple answer would be "yes", but there is much more to Virginia Woolf than "yes" and "no" answers, as you will find out the more you read about her and her works.
As a genre-bending production that combines the basics of essay, discourse, self-analysis, and autobiography, "A Room of One's Own" does show plenty of the stream of consciousness narrative in that it features:
Characters are consistently talking to themselves. We are not the intended audience. We really are the secondary listeners.
Woolf is determined to make a point throughout this essay regarding her conviction that there should be equality for men and women. She does this consistently.
- Non sequential (non linear) narrative
Can we really categorize the story by beginning, middle or end? We cannot. It is all over the place, sometimes with some parts of the narrative showing more organization than others.
There is literary license to shift from one theme to another, or even from one mood to another, as well.
Again, we are not being asked to sit down and read, or sit and listen. The narrator will do it, regardless.
In other words, Virginia is not speaking to us directly, nor is she trying to convince us of anything. In literary works that feature this narrative style, the characters who do the talking seem more like they are making an attempt at convincing themselves of what they are saying.
It is no different than speaking out a manifesto where an idea is exposed and evidence is given to support the main thesis, hence, the "essay" traits also evident in this work.
Whether it is as Mary Beton, Mary Seton, Mary Carmichael, or even as Judith Shakespeare, the reader gets the gist that Virginia Woolf is speaking about a topic that is very dear and strong to her: gender equality in literature. As such, she uses a variety of foci to make her point, present her ideas, and perhaps elicit a sense of agreement in the audience, to justify her point.