Can a food allergy result in skin rashes but not digestive discomfort?

Yes. Food allergy can present in several different organ systems. Digestive symptoms is only one. Skin rashes such as hives or eczema may be a presentation of food allergy without any other symptoms.
Possible. Celiac disease, intolerance to gluten in wheat, is also associated with dermatitis herpetiformis and skin rash may be present in a person with mild form of intestinal disease.