Your wife, should see a specialist like a Allergist/ASthma doctor. I would suggest that your wife have some Allergy Testing Done by a specialist. If your wife hasn't been through this already Have all the necessary breathing test there are, Spirometery, Peak fLOW, there are several other ones. Acid Reflex could be a possiblity. Does your wife experience frequent Sinus Infections? they can cause Post Nasal Drip that has a terrible cough with it. Coughing to the point of vomitting. As far as Asthma Goes not all people with Asthma Wheeze. There are several different syptoms of Asthma, every Asthmatic is different they don't always have the same syptoms.
Asthma Symptoms: Coughing,Wheezing, Chest Tightness, Rapid Breathing, Shortness of Breath. Actually, sometimes Coughing is the only syptom of ASTHMA. Your wife, may have to experience with many different types of Asthma medication before there is an improvement. I strongly recommend a Asthma Controller Medication called Advair Diskus.
I currently take that medication, and It has saved my life not kidding. Nebulizer at home might just benefit your wife. Coughing constantly and vomitting like your saying your wife does has to do with inflammation in your wife's lungs. Whether it be an viral infection or A bacterial infection. Things that you were explaining does sound like Asthma. I also have Asthma. I would definetly see a Specialist. Hope, this helps some for you. Let us know your wife progress. Have a nice day!