Most of the elements of Southern Gothic are present and correct in "Why I Live at the P.O." For one thing, Sister lives at the post office in order to escape her eccentric family, a common feature of the genre. Then there are the somewhat grotesque portrayals of her family, warts and all. These strange characters are depicted by Sister as larger-than-life to the point of being cartoonish. And the situations that Sister recounts are suitably absurd and overblown, such as the bizarre incident in which Uncle Rondo wears one of Stella-Rondo's flesh-colored kimonos. All this is grist to the Southern Gothic mill.
Sister's status as an unreliable narrator is also characteristic of Southern Gothic literature. The whole story is told from her point of view, and it's a fair bet that, at the very least, she's greatly exaggerating the course of events. Sister is so singularly lacking in self-awareness that she doesn't realize that the vicious, mean-spirited picture she paints of her family members says a lot more about her than it does about them. This is Southern Gothic to a tee.
I'm more familiar with the term "Southern gothic" than I am with the term "Southern grotesque," but I suspect that they're connected terms. One commonly discussed element of Southern gothic fiction is the grotesque.
Eudora Welty's short story "Why I Live at the P.O." can certainly be understood as an example of Southern gothic fiction. Nearly all of the story (everything but the ending, from what I recall) takes place in the family estate, a place of faded prosperity and well established authority that closely mirrors "the ruined castles of nineteenth-century Gothic romances, with both symbolically signalling the end of an era."
See the links below for further discussion of this topic.