Westies are notorious for their sensitivities (they fall into the "little white dog" category). Is the food a new food that you have just switched him to? Does he scarf his food down really fast? Are his portion sizes too large? Withhold food for 24 hours (water is fine). Then after 24 hours try feeding him BOILED lean (with grease drained off of it) hamburger (or BOILED chicken) and rice. The mixture should be 1/3 of the meat choice and 2/3 of the rice. Feed small, frequent meals (roughly 4 small meals each day). Feed that exclusively for 5-7 days, then GRADUALLY start mixing his regular food back in. If he begins vomiting again he should be seen by his veterinarian. His veterinarian may also recommend a hypoallergenic diet (NOT a lamb formula as this is no longer a hypoallergenic meat source). If it is a new diet, it's probably a good idea to go back to the last diet he had that didn't make him sick (do this after 5-7 days on the hamburger/rice diet). If he eats too fast, there are food bowls that you can get at pet stores that slow dogs down while they are eating. If he is getting too much to eat, try to divy up his portions into a few meals a day rather than one or two. Obviously the above recommendation is only if your little one isn't lethargic, blood in the vomit, perfuse diarrhea, or anorexic. If he is....see your veterinarian INSTEAD of the above recommendations. Good Luck!
