Do vaccines cause brain damage or encephalopathies?

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).
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.