My 4 yo son has chronic eczema, high IgE antibodies & double the normal limit of eosinophils. Is this caused by a food allergy? I'm thinking dairy...

Could be. Elevated ige and eosinophils are common in children with eczema. This can indicate allergies to either foods or airborne allergens. See a board certified allergist for evaluation.
Allergy and eczema. Allergy is the most common trigger for childhood eczema, though in many children it is difficult to identify an allergic cause. Food allergy may be a trigger, as well as pet or dust mite exposure.
to see allergist. Speak to your pediatrician , ( may be has already in mind to refer )it may be " atopic dermatitis ', an immunological problem , need allergy/ immunology specialist , some one in your family may have it also , may be manifesting as asthma etc rather than cutaneous form, .