Is it possible for my toddler to be allergic to dogs even though her only symptom is a rash from being licked?

Very possible. It is very possible for your child to be allergic to dogs, the protiens in the dog saliva may react to your childs skin it is best to consult an allergist for proper testing.
Is So,Probably Not. If they react to bring licked with no other symptoms at all, it suggests a local irritation from a chemical in the saliva. A reaction, but probably not a true allergy.