I believe the Emily's relationship with her sister Susan is the only one discussed in the story because it is the one most significant to her. The narrator, Emily and Susan's mother, says that, while
"there were conflicts between the others too, each one human, needing, demanding, hurting, taking - only between Emily and Susan, no, Emily toward Susan (was there) that corroding resentment."
Emily is the first child, and Susan the second. Emily had had little enough love and attention from her mother to begin with, and the addition of Susan only left that much less of her mother's time available to her. Also, since Susan was the next child born after Emily, it was with her that Emily had to first learn sibling interaction, and competition. Because of their contrasting natures, Emily found herself sadly wanting.
To make matters worse, Susan's birth coincided with Emily being very sick. Emily, who needed her mother's care desperately during that time, was actually forbidden to come near her new sister and mother for the first week the baby was home to avoid contagion. It is difficult to imagine the desolation Emily must have felt at this situation, already knowing only that she wanted her mother but was denied her because of the baby, and then came the final insult, when she was sent away to a convalescent home. The narrator explores Emily's relationship with Susan because the fact of Susan had a definite impact on Emily's life. She traces many of Emily's difficulties in adjusting to life directly to the birth of Susan.
Emily is understandably insecure about herself, and Susan,
"golden-and-curly-haired and chubby, quick and articulate and assured, (is) everything in appearance and manner Emily (is) not."
While the children that came after Susan must certainly had some influence on Emily's development, it is clear that none of them, at least in the narrator's eyes, had the impact that Susan did.