Do vaccines cause brain damage or encephalopathies?

No. Before we had the current DTaP vaccine in the mid 90's many kids had fever & an occasional one had a seizure (with fever). The only benefit of the vaccine hater activity was enabling the "clean up " of early vaccines to the point they are truly safe & effective. Over the years. Vaccine haters claimed it as a cause of crib death, CP & now autism/brain injury. Over time each claim is refuted.
Brain damage? No. There is no definite evidence that any vaccine directly damages brain. Encephalopathy had been noted very infrequently with DPT containing whole-cell pertussis vaccine (not in use in us).