First off let me tell you that for puppies, you should stay away from rawhides and pig ears. These can be swallowed in large pieces and lead to intestinal obstructions. Symptoms of an intestinal obstruction would be vomiting, an inability to defecate and loss of appetite. You can read more on this here:
If the chew stick was rawhide, then many vets will recommend giving it a day to see if it softens and moves through the system on its own. For other non food items that might be ingested, it is recommended that the puppy be seen immediately if you suspect an obstruction. This is because the longer a blockage exists, the higher the chance of permanent injury to the intestines.
That said, your pup may have initially vomited due to ingestion of fecal matter. The subsequent vomiting may be due to an empty stomach and excess bile. A whitish or yellowiish vomit is sometimes due to excess bile. We frequently see this in dogs that are not eating or first thing in the morning after a dog has not eaten in some times. You can give your pup some pepcid. Read about Pepcid dosages and usage information here:
Give the pepcid about 45 minutes to work and then offer a small portion of boiled chicken and rice. If the pup does well, then you can slowly add regular food to the mixture and decrease the rice mixture until the pup is back on regular food.
We do recommend withholding food if a dog is vomiting, but I would not do that with an isolated instance of vomiting. Dogs vomit for many different reasons including parasites, an unusual treat, severe protozoan infections might cause vomiting, excess bile, poisoning and pancreatitis to name some. In some cases fasting is recommended such as pancreatitis. In others, switching to a bland diet like the chicken and rice might help. If your pup continues to vomit, put a drop of karo or pancake syrup on your pup's tongue to keep the blood sugar levels up and prevent hypoglycemia. You can read about that here:
If your pup does not seem to be improving then I would have your pup seen. He should be protected against diseases like Parvovirus, but with repeated vomiting he can become dehydrated quickly and might need rehdration. I hope this information is helpful to you.