I agree with looking into allergy testing.
My in laws dog is a german sheppard, and has always had digestive problems. The vet put him on a duck and potato food because it was supposed to be easy to digest. But since it wasn't getting better, they had him tested for allergies and it turns out, he's allergic to duck (among a few other things).
I have my dogs on Orijen 6-fish formula. The only protein source is from 6 different kinds of fish, no other meat, so technically it's vegetarian
. Plus it's completely grain free, and they absolutely love the taste/