"But I can't "realize it is supported by real data" when it clearly is not"
But this is somewhat my point. When you say "clearly is not", have you really tried to find it? or assume it is not there. You acknowledge you have seen some data such as the toxic ingredients. So in this case you could have a discussion around the interpretation of the data etc.
My point is that we shouldn't assume the other side are always ignorant, uneducated or just following Jenny McCarthy etc. You will find plenty of doctors, neurosurgeons etc that also have issues with vaccines. Dr. Wakefield became famous for the autism issue; however, you should be aware that since then there are many educated doctors that find some issue with vaccines. Of course being a doctor doesn’t make you automatically correct, it is just once again being aware that the person you are debating may have what appears to be credible information.
It would be best to ask where the person is coming from in a non hostile way. Just ask, so what have you discovered and what led you to your decisions for example. That is friendly and opens up conversation. From then you can discern if this person made decisions based on credible sources ( but maybe not accurate ) or they have not actually educated themselves.
When I say credible sources, I mean to say at least they are referencing other doctors, scientists, studies vs some blog. Credible in this instance doesn't necessarily mean correct or wrong, but acknowledging they did their research will allow conversation to continue. Then you can share counter research etc as conversation progresses.
Just be aware, within the scientific community, these topic still are being researched and it is not as if they are completely closed. For example investigations still continue into mercury and autism. Being aware of such is always a plus when having such discussions as you will appear to at least know something about where they are coming from. http://www.ncbi.nlm.nih.gov/pubmed/24995277