I'm skeptical that there's actual evidence of severe adverse reactions (aside from the occasional allergic reaction). "I had a vaccine and then this bad thing happened to me," is not an indication that the vaccine caused the bad thing. It might have, but the severe reactions have been so incredibly rare that there's really no evidence of a causal link, as near as I can tell.
But what you are asking for here is a far, far higher barrier to obtaining a vaccination than is asked for for most any other medical procedure or remedy. The real information is, "This will protect your child, and the population as a whole, from serious diseases. It most likely won't cause any issues. Your child may have minor cold symptoms for a bit, which means the vaccine is working."
The CDC's page is informative here: http://www.cdc.gov/vaccines/va...
Note that under the "severe" reactions is usually the disclaimer that they can't actually be sure this reaction is caused by the vaccine. I'd be willing to bet that disclaimer should really be expanded to encompass every vaccine on the list, aside from the allergic reactions.