A dog vomiting bile can be common, however, if a dog is vomiting bile and shaking you most likely have a dog with an illness that will require some intervention to resolve. There are a few things that you can try at home to see if it helps your dog prior to seeking help from a veterinarian.
First Consider the Feeding Schedule
Sometimes when dogs get an upset stomach simply keeping the dog from ingesting food and water for a few hours after they have vomited, will resolve the symptoms. However, oftentimes vomiting yellow bile is a sign that a dog hasn't eaten in a long time, or there has been too long a period of time between meals. If the bile is yellow and frothy, the owner should try feeding small meals throughout the day to see if that solves the problem/
Change the Type of Food
Some dogs have more sensitive stomachs than others. Changing the dog's diet can help in some cases.
Have Your Dog Tested for Allergies
Dogs can have environmental allergies. Having your dog tested for allergies is one way to rule out some causes of the symptoms. Allergy testing is also a way to possibly discover the source of the symptoms so you can remove the allergen from the dog's environment.
It is important that you make sure your dog does not vomit so much that it becomes dehydrated. While withholding food and water for a few hours is a good way to see if you can stop the vomiting, you don't want to deprive them of water for too long. Most dogs will vomit from time to time, but if this symptoms are more chronic exploring other possible causes of the symptoms with your veterinarian will be an important next step.