The protagonist of the story is the main character around whom most, if not the plot, occurs. This character is the most affected by the consequences of the story.
Considering this, Framton Nuttel is certainly the character who is most affected by the actions concocted by Vera who, by default, would be his antagonist.
An antagonist is described as a "rival", or "opponent" of the protagonist. This description would fit Vera perfectly, as she deliberately plots an action knowing how it would affect the feeble character of the nervous wreck that is Framton Nuttel.
Vera's actions are foreshadowed at the very beginning of the story
"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."
This suggests that Vera has a tendency to terrorize or annoy people. It also means that she thinks ahead, for a lady who is "self-possessed" at a young age must know what she is doing.
Moreover, Vera keeps her cool even at the end after having caused Mr. Nuttel to run away from Mrs. Sappleton's house. She even lies, giving false information about what may have caused for Nuttel to act that way.
I expect it was the spaniel," said the niece calmly; "he told me he had a horror of dogs. He was once hunted into a cemetery somewhere on the banks of the Ganges by a pack of pariah dogs, and had to spend the night in a newly dug grave with the creatures snarling and grinning and foaming just above him. Enough to make anyone their nerve."
Romance at short notice was her speciality.
It is interesting to note how, like the last quote states, Vera is capable of coming up with all these stories just for the sake of telling them and cause drama. This is a quality inherent to her, which causes pain in the protagonist. Hence, she is the epitome of the antagonist.