I hate to hedge in answering a question, but this is a question that truly could be answered either way.  I am going to list some of the elements of character development that argue for a flat character and some of the elements that argue for a round character.  You will be able to decide which you find more persuasive, as well as possibly thinking about other elements that I have not thought about.

August could be said to be a flat character for at least a few reasons. First, there is little evidence of "growth" in her over the course of the story.  She is presented as a fully formed adult, and there are no changes in her, as opposed to June, for example, who decides to open up emotionally and marry and who manages to get past her prejudices against white people.  August presents as perfect, really, and she remains perfect throughout.  And that brings me to the second reason she could be said to be a flat character, which is that she exists in the book as a kind of center for all the characters, a sun around which they all revolve, a touchstone for Lily, her symbol of mother, and the queen bee in the story.  Third, because the story is narrated by Lily, we do not have all that much insight into August, except what Lily has chosen to share. This is Lily's journey, and much of it is solipsistic, so we do not have a great sense of August's struggles and growth, not all that much evidence of the inner workings of August's mind.

To the degree that August is a round character, it is in the evidence of her back story and the qualities that she does display in the novel.  This is a woman who has struggled to be who she is, a journey of pain, prejudice, and loss. She has been through a great deal, and this has made her strong and loving.  We see that she is patient, waiting for Lily to come to her.  We see that she is wise and compassionate, understanding the needs of those around her, always saying and doing just the right thing.  So, the presentation of her qualities, her character, her voice, and her actions could reasonably be said to make her a round character.

I think in much of literature, we are likely to find evidence of both flatness and roundedness in characters.  It is often simply a question of degree. Authors probably do not set out with this in their minds, and it is up to us to decide how to impose these contructs upon their characters.