First of all, dogs are carnivores, and any commercial kibble that calls itself vegetarian is a just trying to appeal to human diet trends and not addressing the diety requirement of dogs. And, it's heavily processed filler, not food.
If you are trying to eliminate possible food alergies, simplify the diet to one specific type of food at a time. Try feeding only raw chicken for a week or two. The itching will either get worse or better. If it gets better, try beef, turkey, lamb, duck, organs, green tripe... guess what? You're feeding RAW!
If one particular animal makes it worse, don't feed that animal, and continue feeding RAW.
Either way, the answer is to do a systematic switch to a RAW diet. Easy, inexpensive, and the best thing you can do for your dogs health.