The crux of most anti-vax opinion is that it causes autism - because there was an observed increase in vaccinations and autism diagnoses...which was irrelevant. There was a landmark study that showed more than mere correlation, but it was found to have used falsified data and troves of research has tried to replicate it and found no such causal association.
I suppose the movement may have developed some other claims since then, but I'm not aware of them.
Over-vaccinating...sheesh, that doesn't even make sense. Over-medicating? Sure. Vaccinations...nope, I can't even fathom the logic