I think they do. When you only see post about how vaccine cause autism, anectode about this and that person and the diseases they got from the vaccine and that on top of that the vaccine doesn't even prevent the disease it was designed against, then it becomes reasonable to become antivax.

And if effectively Facebook knowingly choose, through their algorithm parameters selection, to promote this material as it increases engagement more than reasonable content, then yes, I think they should at least be partly held responsible for the harm caused by the anti vaccine movement.

And this is only one example.