Could iron deficiency cause shortness of breath?

Yes: Iron deficiency causes anemia or low red blood cell counts. Red blood cells carry oxygen so lower numbers can result in shortness of breath.

Indirectly...: If the iron deficiency is severe enough, iron deficiency anemia will result. With this anemia, there is less hemoglobin to carry the oxygen molecules to the tissues and this will cause shortness of breath. See your doctor for diagnosis and treatment of iron deficiency anemia.