Food allergies typically produce red itchy feet (extremities) and ear infections, but not normally the sort of red itchy bumps in the armpits and haieloss you're describing. IF allergy related at all, I'd be more inclined to suspect a contact allergy.
Another possibility though, is sarcoptic mange - for which the classic symptoms are very itchy skin, hairloss in the armpits, spreading to the rest of the abdomen, and red bumps that may become infected (especially if the dog scratches or licks - which inevitably they do). You'd need to see a vet for diagnosis though, and for treatment in the event it is sarcoptic mange. Bear in mind that sarcoptic mange is infectious, so if this is a possibility, you'd need to keep your dog away from others till treated. More info about it here - if this sounds like your problem, then I'd suggest seeing a vet right away:
Canine Scabies: Sarcoptic Mange in Dogs