It is possible your dog may have caught an upper respiratory illness such as kennel cough or possibly canine influenza. A heart condition such as heartworms can also cause a cough, which will need to be checked if your dog is not on heartworm preventative or an older dog. Kennel cough is normally contracted when a dog has been boarded or kenneled or around a large number of dogs such as at a dog show, dog park or pet store. Here is a website with more information on kennel cough. http://www.thepetcenter.com/gen/kenc.html
Canine influenza is now becoming more prevalent and like it sounds it is a canine flu. Here is an excellent site on it.
You will want to monitor your dogs condition looking for colored discharge from the nose or eyes, a productive cough (coughs stuff up), stops eating or lethargy. These are signs of a possible bacterial infection as well and my require antibiotics. If your dog appears to be having a difficult time breathing, you will need to see your Vet as some dogs dog get really sick with canine influenza and need support to recover.
To help your dog breathe easier you can run a NON-medicated humidifier in the room your dog is in, or sit in a steamed up bathroom with your dog to help keep the mucous moving. Robitussin DM at a 1/4 teaspoon per 5 pounds can be used to control the cough. Dosages can be obtained at this website. http://www.petplace.com/drug-library/dextromethorphan-robitussin-dm/page1.aspx
I believe the coughing is causing the vomiting, but to be on the safe side, there are a couple of things that can be done for it if it is not related to the cough. You can give your dog some pepcid or Zantac. Read about Pepcid dosages and usage information here: http://www.petplace.com/drug-library/famotidine-pepcid/page1.aspx
Zantac can be given to a dog at .25 to 1mg per pound every 8-12 hours. Read about usage here:
I'd start feeding your dog smaller meals 4 times a day taking up the food bowl afterwards. This way your dog will have food in its belly on a regular basis throughout the day. This will cut down on excess bile. I'd also add a spoonful of plain yogurt to your dog’s food 2-3 times a week to add good bacteria to your dog’s stomach to aid in digestion as well. The bland diet of boiled chicken and rice can be used for a few days. If your dog does fine, you can slowly reintroduce your dog’s regular food into the chicken and rice mixture until it is all regular food. If these things do not work for your dog, then your dog will need to go to the vet and have tests run to determine if a more serious condition is present.
The following site goes over canine coughs.
I hope you find this information helpful.