The eNotes Guide to Literary Terms defines "protagonist" as follows:
Protagonist: the leading character of a drama, novel, etc. This is not always the hero, but is always the principal and central character whose rival is the antagonist.
In Sarah Orne Jewett's short story "Miss Esther's Guest," Esther Porley is the protagonist, around whose thoughts and feelings the action and resolution of this gentle tale are centred. The story opens on the image of Miss Esther regarding herself in a mirror, so that the reader can perceive her as she perceives herself: an old-fashioned, kindly, slightly nervous middle-aged woman. She is frugal, wearing a shawl that once belonged to her mother, and gloves that are past their best. She is also very proper, observing a kind of formality in her dress and manners that is perhaps unnecessary in the friendly and close-knit village of Daleham. She is both well-respected and deeply involved in the community, as Jewett describes:
Every one in Daleham knew the good woman; she was one of the unchanging persons, always to be found in her place, and always pleased and friendly and ready to take an interest in old and young.
These traits of Miss Esther's—frugality, good manners, and community feeling—are traditional virtues of New England's culture, which has its roots in Puritan simplicity. Miss Esther, at sixty-four, is a pillar of her little village. She is also, however, very lonely. Having lived with her mother her entire life, Miss Esther is an "old maid," and since her mother's death three years previously, she has lived completely by herself:
She was made very sorrowful by her loneliness, but she never could be persuaded to take anybody to board: she could not bear to think of any one's taking her mother's place.
The story concerns Miss Esther's decision to take in a guest to alleviate her loneliness. She "requests" a guest from the Committee for the Country Week, a kind of welfare group which sends city-dwellers on brief vacations to board with people in country villages. The Country Week is seen as an important respite for city-dwellers from the noise and pollution of the city, and anyone can sign up to be a host through their local parish. Miss Esther signs up and requests that the guest be an elderly woman, similar to her mother.
The guest she receives is in fact a man, Mr. Rill. It could be said that Mr. Rill is the antagonist, by virtue of being an unexpected presence in Miss Esther's life, but this is a severe stretch of the term. According to the eNotes Guide to Literary Terms:
Antagonist: the character who strives against another main character. This character opposes the hero or protagonist in drama. The term is also used to describe one who contends with or opposes another in a fight, conflict, or battle of wills. In literature, this is the principal opponent or foil of the main character and is considered the villain unless the protagonist is a villain; in that case, the antagonist is the hero.
Mr. Rill is the furthest thing from a villain and to consider him a "rival" to Miss Esther is a bit ridiculous. Jewett describes him as "a sensible old person," gentle and reserved in manner. He, like Miss Esther, has never been married, having lived with his sister for many years until her passing. He, like Miss Esther, is "very lonely," and like Miss Esther, he is rather anxious about spending time with a stranger. Miss Esther has a shock when she realizes that her guest is a man rather than the woman she requested, and there is a moment of hesitation in which she and Mr. Rill nearly agree to part ways, but then Miss Esther strengthens her resolve and welcomes the guest into her home. The two find they quite enjoy each other's company, and by the end of the story, Mr. Rill has proposed to Miss Esther and is making plans to move to Daleham permanently.
If Mr. Rill is not the antagonist, is there one in the story at all? It's a good question—there are no characters who overtly oppose, "strive with," or otherwise obstruct Miss Esther in the course of the story. The only source of conflict is her own anxiety at doing something new by taking in a guest—anxiety that is briefly sharpened when she realizes her guest is a man. But this anxiety swiftly passes, and it comes from with Miss Esther herself, rather than being imposed upon her by external circumstances. I would therefore venture that there is no antagonist in this short story, only a protagonist.