Can having a milk allergy cause circles to form under the eyes?

Usually no.: True milk allergy is due to allergic antibodies (ige) which causes a dramatic immediate reaction, itchy lips/mouth, throat tightness, nausea, vomitting, abdominal cramps, diarrhea, hives and in addition, stuffy nose and chest tightness. The reaction clears within 24 hours, so chronic circles under the eyes would not be due to milk allergy.

Allergic shiners: Any allergy that causes nasal stuffiness will lead to dark, swollen circles under your eyes. This is called allergic shiners. Blood leaving your nose passes through a network of small veins below your eyes. Since allergy increases nasal blood flow, the increased blood in the veins below your eyes give the skin a bluish color. Plasma may leak out of these veins resulting in edema (swelling).

Less likely: Typically dark circles under the eyes, termed infraorbital darkening or allergic shiners, is due to environmental allergies with nasal congestion and subsequent back of blood in the small blood vessels near the eye. Food allergies are more likely to cause GI symptoms and rashes like hives or flares of allergic eczema (atopic dermatitis).