The problem with this claim is that the link is correlation which is not evidence of causation. Again, another non-scientific claim.
That said, I do support parental rights in medical decision making. The mortality rate from measles has been greatly reduced by improved nutrition and medical advances. I am not a big fan of one size fits everybody programs in any area of life.
I myself did not follow the schedule with my children, and made the multiple trips necessary to space the shots out. In fact I didn’t do the Hep C series for infants until they were teens. Same with the stupid varicella shot. I figured is was better to catch it as a child than a vaccination that provides uncertain immunity for an uncertain period of time.