I think it has to do with the "I'm an intelligent, responsible doctor/scientist/researcher, not a crazy internet conspiracy theorist" feeling among those who push vaccines. While there may be large unvaxed populations out there to compare with the vaxed group, it would be impossible (with current medical and political vaccination pressure) to design and implement a double blind study, which is one of the best types of studies with (supposedly) some of the most reliable results. (A double blind study would randomly assign vax or no vax to individual parents/children, and neither the doctors nor patients would know who had been assigned to what group.) So, if anti-vax groups performed a "low-quality" study, the medical community would dismiss its results as not valid.
If pro-vax/medical folks caried out this "low-quality" study, and its data supported not vax'ing, what then? More studies would have to be performed, but they would need them to be "better quality," which would necessitate double blinding, which would be "unethical." Not to mention that even to design/fund such a study would be akin to admitting that there may be problems with vaccination. No one in the pro-vax camp wants to open that can of worms.
Even with such a study, it would be difficult to "prove" anything either way. What would the study be measuring? How would you determine whether any vaccine-related problems/risks the study may bring to light outweigh the "benefits" of vaccination over the last several decades? We can't now accurately measure how much vaccination can actually be attributed historically to disease decline, although the popular view is that they have been essential in this role. If the study showed that vaccines are in fact detrimental to some fraction (or all) of the population, pro-vaxers would marginalize these results in favor of the greater good that vaccines have supposedly done.
Also, vaccination/disease risk in the US isn't the same as vaccination/disease risk in South Africa, for instance. Assuming a satisfactory study could be done in the US with clear results showing the problems/risks of vaccination for American children, where does that leave the rest of the world? It would open up ethical questions and issues in all sorts of areas, from WHO recommendations to charitable and government agencies who provide free or low-cost vaccinations to the poorest children in Africa. Would they continue? How would we quantify the risks to those children from vaccination in light of the risks they face from contaminated water, insufficient food, endemic disease, etc?
Then of course I also believe that there is a certain element of pride among doctors, medical organizations, the CDC, WHO, etc that prevents them from even seeking any further for answers. What if they have been wrong all these years? Their credibility would crumble. And while I am certainly no proponent of most of these groups in general, I do believe there are very good things that some of them do, in some areas. If the emperor has no clothes in this instance, what impact would it have on the health and well-being of certain people in other arenas (for example, maternal and infant health, breastfeeding, etc)?
I agree that such a study would be interesting. There are just so many other questions and disincentives for either camp to go there.