Can allergies cause chronic coughing?

Yes. Allergies can cause chronic coughing, but a doctor will evaluate to look for other causes. Asthma (some people cough but don't wheeze), gastroesophageal reflux (heartburn), allergic rhinitis (allergies in nose), non-allergic rhinitis (runny nose with negative allergy tests), TB (tuberculosis infection), throat cancer, lung cancer, sinusitis, drug side effects, etc... Can all cause chronic cough.
Yes. Their are many causes, and allergy is one of them.