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.
In the book "Charlotte's Web" the conflict in the story is that Wilber is a farm pig who will be slaughtered for food. Charlotte, the spider and Wilber becomes friends. Charlotte's efforts to save Wilber are driven by the conflict, Wilber's impending death. It is because of this that the story line moves forward as Charlotte tries through various means to save Wilber. In the end of the story her final sacrifice is her life as she uses the last of her energy to spin a final message assuring that Wilber will be safe. Wilber in turn is left the job of caring and befriending Charlotte's young off spring.