We got word back from the vet on why Gizmo died so suddenly. Apparently for a long time now Gizmo had been dealing with a bad build up of some kind of bacteria in her intestines. According to the vet, there are countless things that can cause this to happen to dogs, but they have no idea what caused it in this case. This is partly the cause of her sensitive stomach and explains why she vomited so frequently throughout her short life. Apparently, the last time she got sick she must have aspirated (inhaled) some of her vomit and that caused pneumonia. The combination of the pneumonia and the intestinal bacteria is what got her and the vet agreed that when this sort of thing does happen, it happens quickly. He did say that the type of bacteria wasn't contagious so we don't have to worry about our other dogs.
The lesson here is that if you have a dog with a "sensitive stomach", take it seriously. The fact is that if your dog periodically throws up for no apparent reason might mean that you should take it to the vet.
-- Matt Ranlett