More important than the cause is the cure. I think probiotics would be very helpful. VSL#3 contains 450 Billion CFU per packet, which is higher than any other probiotic brand that I know of. The unflavored can be mixed with food such as yogurt. I also add one scoop of Prebiotin prebiotic fiber to the yogurt, which helps the probiotics stay alive.
I think the vax connection is a possibility if the constipation turns out to be autoimmune; celiac disease (which is an autoimmune reaction to gluten, found in wheat, rye, and barley) causes chronic diarrhea for most, but chrnoic constipation for others, or even alternating diarrhea and constipation.
Celiac disease, like other autoimmune disorders, can be triggered by vaccines.
A high percentage of autistic children have celiac disease, or something extremely similar called "gluten intolerance." Some doctors feel that gluten intolerance is simply early-stage celiac disease, before there is enough intestinal damage to definitively diagnose celiac disease. Others believe that they are two separate disorders, that happen to cause exactly the same symptoms.
Because celiac disease causes intestinal malabsorption, vitamin deficiencies are usually the result, even if the patient has a healthy diet and takes vitamin supplements.
Autism (and ADHD, which is considered to be on the autism spectrum as well) seems to have a very, very high rate of vitamin deficiencies, which makes sense if there is anything causing malabsorption.
You might try searching sites like www.celiac.com and www.csaceliacs.org for information on autism/ADD/ADHD and gluten. Many parents report that their children's symptoms drastically improved after starting a gluten-free diet (most of those also took their kids off all dairy products as well, at least temporarily).
Other good and interesting sources are Special Diets for Special Kids by Lisa Lewis and http://www.karynseroussi.com/home.html