If puppy ate left over cat toy, then you should have seen bits and pieces in the initial vomit.

If puppy likes to eat pre-digested cat food (feces) then one would expect diarrhea over nausea.

Possibly kitty gets off free on this one.

There are several possibilities ranging from food allergies, to metabolic disease and disorders. You definitely need to check in with a Vet, and get a blood panel run.