What could cause daily morning cough that leads to vomiting?

Maybe LERD?: Exposure of the vocal cords to stomach acid & pepsin will result in inflammation of the vocal cords (as is also often seen with acid regurgitation) = lerd. Acute laryngitis may result, but watch out for chronic hoarseness, throat clearing, airway spasms, cough, bronchitis, an asthma-type presentation, & even a higher risk of throat cancer. May however not be reflux so please get yourself checked.

Possibilities...: Possible diagnoses include gerd, sinus disease or a combination of factors. See your doctor as there are treatments that are effective for these conditions.