I would say that the protagonist in the story is Charlotte the spider while the antagonist is the farmer, Homer Zuckerman.
The conflict between them is that Homer wants to kill Wilbur while Charlotte is going to do her best to keep him alive with her web writings that extol his virtues.
The plot is driven by this conflict because the main plot line follows Charlotte's efforts to keep Wilbur from being killed. These efforts continue up to the time when he is brought to the fair and wins the prize, thus ensuring his survival. At that point, Charlotte's work is done and she can die in peace.