This is a legiimate question but a hard one to answer. There are only three important characters in the story. It seems pretty obvious that Vera has to be the protagonist because she is the instigator of everything else that happens. If that is the case, then the antagonist has to be either Framton Nuttel or Mrs. Sappleton. For any story to be interesting, it has to be dramatic, and for any story to be dramatic, there has to be conflict. There does not appear to be any conflict at all between Framton and Vera, and therefore I suggest that the conflict has to be between Vera and her aunt.
I suggest that Mrs. Sappleton is the antagonist. She is trying to teach Vera to be a gracious hostess, and she has taken Framton's visit as an opportunity to let Vera practice her hostessing. Vera greets Framton as follows:
"My aunt will be down presently, Mr. Nuttel," said a very self-possessed young lady of fifteen; "in the meantime you must try and put up with me."
Vera must be an angry young lady. She resents being used by her aunt as a surrogate hostess. She does not want to grow up to be a woman like her giddy aunt. Vera resents being confined to the house because she is a female, while the three men go off to shoot birds and have a lark. She takes out her resentment on poor Framton Nuttel, who just happens to be in the wrong place at the wrong time and is neither the protagonist nor the antagonist. Framton might be said to be the "MacGuffin," the bone of contention between protagonist and antagonist.
Mrs. Sappleton wants Vera to play the gracious hostess and make Framton feel comfortable and welcome. Therefore, Vera does just the opposite and makes him feel so uncomfortable that he finally leaps up and runs out of the house in a panic. Her aunt wants a peaceful, quiet, civilized, stable, routine existence. Vera hides her true feelings but does her best to generate excitement, amusement, and uproar. She is using Framton to get back at her aunt for keeping her a prisoner and trying to mold her into another a female doormat like herself.
Vera is clever. First she finds out that Framton is a stranger in the region. She makes her story brief because she knows exactly how long she will have before her aunt appears; and she knows exactly, from boring experience, what her aunt will talk about when she does appear.
"Poor dear aunt, she has often told me how they went out, her husband with his white waterproof coat over his arm, and Ronnie, her youngest brother, singing 'Bertie, why do you bound?' as he always did to tease her, because it got on her nerves. Do you know, sometimes on still, quiet evenings like this, I almost get a creepy feeling that they will all walk in through that window - "
Bertie's singing probably got on Vera's nerves rather than on her aunt's. Vera has seen these scenes and heard her aunt's mindless drivel so many times that she is going a little crazy. She knows that the three hunters will shortly appear with the little spaniel and that Bertie will begin singing "I said, Bertie, why do you bound?" She knows her aunt will be looking forward to their return and talking about the only subject that ever gets talked about in this remote, benighted country household--the shooting of birds.
We can imagine that there is an ongoing contest between Vera and her aunt. Vera feels trapped, suffocated, tied to a Procrustean bed. She wins this little victory in her rebellion, but the reader feels that the contest will go on and on.