The only difference between sample groups was a "priming" article to be read, it seems to me that the study is showing that reading a single article has minimal effect on the average person's opinion, not that people aren't scared of vaccines.
Part of it showed that 25% of subjects believes the autism meme, with lesser percentages on the less prevalent allegorical side effects of vaccines, showing that it's still rather entrenched in society.
Presumably most of these subjects were members of urban or suburban populations. As population density grows, the smaller the percentage needed to break herd immunity (exponentially more disease vectors).
If 25% of parents in relatively well-off populations believe the myth, Dollars to donuts the myth is even more prevalent in rural populations.
Nationwide percentage stats are too broad to determine risk, sample subjects need to be examined at a higher population resolution to tell how accurate the premise is, compare a heat graph of childhood vaccination to believe in anti-vax memes, and my guess is they'd overlap strongly with other socio-economic county heatmaps of the USA, like religiousity, average income, or obesity stats.