My heart goes out to you and your son. This sounds SO miserable for him!
It sounds EXACTLY like what I went through with my daughter before my Naturopath diagnosed her with Celiac disease. I recommend Naturopaths over MDs for food allergies. They treat a "borderline" illness as an illness, not as NOT an illness. The terrible constipation, and the other symptoms you described are exactly the symptoms my daughter had until we cut out wheat. She still has some constipation, but not like before. Something in the digestion is not right - you need to have allergy tests to determine what. Wheat gluten can strip the celia right off the intestines causing the body to no longer absorb food. As a result, the children (like my daughter) end up underweight and TERRIBLY constipated. This has also caused seizures for my daughter and if I hadn't taken her off the wheat, it would have ended for her in autism. This is a VERY SERIOUS food allergy.
You said you have done "food elimination". The question is, how long? For Celiac Disease, you have to actually have them off the wheat for about a month - 3 months to see a turn around becuase the celia has to grow back before you can see a difference in the diet.
I pray and hope for your sake it's not Celiac disease, but that is one of the most destructive diseases to the digestive tract. Vaccinations don't necessarily cause it, it could be caused by the host of genetically modified foods that are out there, it can be genetic, it can be due to a lot of different factors, none of which have anything to do with what the parents did or did not do.
My daughter didn't have her first vaccines until she was 6 mos.