What is the principal social institution against which the narrator of The Yellow Wallpaper struggles?
I don't understand the question, and would like to know what it is trying to ask so I can answer the question.
Before being able to answer the question, one does need to understand the question posed to them. I assume that the barrier you are facing is the phrase "social institution."
A social institution, according to the Stanford Encyclopedia of Philosophy, is
complex social forms that reproduce themselves such as governments, the family, human languages, universities, hospitals, business corporations, and legal systems.
The unnamed speaker (the narrator) of Gilman's short story, The Yellow Wallpaper, is not fighting against the government, language barriers, a business, or the legal system. Instead, she is fighting against the family. Her husband, recognizing a mental illness, requires her to stay in her bedroom. She is kept away from her child (based upon this, one could assume that she is suffering from Postpartum Depression). Therefore, outside of fighting herself (which would identified as internal conflict), she is fighting against her family.