Is wheat allergy genetic?

Yes: it typically presents as a food allergy which involves immunoglobulin E and mast cells. There is a genetic component to it. IgE testing might miss some of the many components of wheat that can cause allergy, so elimination of wheat and improvement of symptoms is advised.

Yes and no: The tendency to become allergic does run in families. However, you do not inherit the allergies of a parent like you would eye color or other similar traits. Those who develop celiac disease ( of which wheat is a trigger) most of those affected fall into certain inherited tissue types, but not all. Not all with those tissue types develop celiac disease.