Food allergies can cause worse problems than just an itchy rash. Dairy can cause hearing loss, for one thing. I could have been spared a lot of misery and life-long problems if my mom had taken food allergies seriously. I feel that was her job. I was in my 40s and had seem many docs for skin rashes and other ailments that they thought were all in my head, when a surgeon figured out I had wheat allergy (Celiac disease). I am also sensitive to dairy.
My boyfriend has fruit allergies and problems with eggs, so it can be a wide range of foods.
Eggs aren't dairy, you know. Best way to tell if it's dairy is to stop feeding dairy products and see what happens. As for your question, "home" remedies differ depending on what is wrong, so you still need a diagnosis. You need to see a doctor and probably a dermatologist to see what's up, and then you can investigate alternative remedies -- nobody forces you to do what the doctor prescribes.
I suggest you take her to a Pediatric Allergist and have her tested. I'm guessing it's Eczema
Don't let her be uncomfortable!