Why does "I Stand Here Ironing" only discuss Emily's relationship with her sister Susan?

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.

What is the theme of Susan and Emily's relationship in I Stand Here Ironing?

Emily is the oldest daughter in the short story, I Stand Here Ironing by Tillie Olsen. Life has been hard on her mother who, only nineteen herself, was left with Emily to care for when Emily's father decided to leave and "wrote in his good-bye note" that he could no longer cope with their life of "want." The mother relates how she was unable to enjoy Emily's younger years due to work demands and only when she remarries do things improve "and I think perhaps it was a better time."

However, when Emily is five, Susan is born and Emily is already coming down with a fever which will culminate in Measles. Her first associations with Susan are all negative as she is not allowed near the baby and is sent away to convalesce for an extended period. The reader can sense the developing relationship already and, for Emily who already has a compromised relationship with her mother, Susan will become symbolic of a difficult time for her, in a place where "They don't like you to love anybody here." and where "There never was a star"- indicative of Emily's failure to thrive. 

The mother is aware of "that poisonous feeling between them, that terrible balancing of hurts and needs." The mother's inadequacies cannot stop the "corroding resentment" and the reader can recognize how the mother must have compared the sisters with Susan being favored as she is "everything in appearance and manner Emily was not."  Susan repeats Emily's jokes and is applauded whilst Emily "sat silent,' telling her mother later but the mother not doing anything about it. 

All of this supports the theme of sibling rivalry and how, unchecked, it may develop into a bitter separation and cause a rift too large to repair as adults, especially in Emily's case, who is already battling the unresolved issues in her bond with her mother.