Based on Miss Franny Block's reaction of being so terrified that she falls on the floor and hides behind her desk when she sees Winn-Dixie looking in the library window, the most reasonable inference is that:
She is terrified of all animals but especially dogs.
While other explanations, such as suffering from long-term trauma or having a medical condition, could be considered, the context suggests a specific fear of animals rather than a medical emergency or general clumsiness.