Your son sounds exactly like mine!
You need to get a referral from your family doctor to see an Allergist. My son was 3 before we figured out his skin rashes and constant "colds" were actually allergies to some of the things in our home. We dealt with endless nights of coughing fits followed by vomiting, runny noses and irritated throats. He had eczema and other weird looking rashes on his arms and legs, got terrible diaper rashes and had very black circles under his eyes. Our family doctor tried several different medicines and tried breathing treatments but nothing seemed to help until we got with the specialist.
It's not likely that it is a food allergy, as it would probably show up right away if it was something he ate. More likely it is something he comes in contact with, it could be anything from your laundry soap to the pollen on the grass in your yard.
We had the testing done and learned that our son is allergic to dust, mold, trees, pets and grass. We had some changes to do to our home, but with the right medication and his regular allergy injections he is a happy, healthy, not itchy little boy. By the way, it is possible he will grow out of these allergies, but it may not be for several years.