Do you have a photo of her back? What concerns me is the possibility of cutaneous vasculitis. This can happen from vaccines. It does not itch to my knowledge, but can look like a burn.
Here is info on it...and if you google it you can view lots of images of what it looks like:
Inflammation of the Skin Blood Vessels in Dogs | petMD
She definitely needs to see a vet and I would mention the above to him. If she has it the treatment is prednisone or another med I cannot think of right now for at least three weeks....possibly longer. Without treatment it can get really bad.
As for food, is she on a puppy food? I have pups on this food that you may shy away from, but I can tell you that I have used it with many pups and they have done very well. At least look at the link before being put off with the name Hill's.
Natural Dog Food - Hill's Ideal Balance Dog Food
I am surprised the vet didn't suggest a prescription food for now such as Hill's ID or Purina EN.