With vaccinations being so universal as every child is required by law to have them it seems it would be very difficult to get a control group of unvaccinated children.
According to the researchers among the study population of 47,155 children of age nine years, 10.9% (5,123) were unvaccinated. Or at least they didn’t have a Medicaid code on record. Maybe possible in a Medicaid cohort but the rate of unvaccinated children in this sample is certainly high compared to other groups and national averages.