Note: Another broader question about anti-vax opinions and political leanings has been marked as a duplicate of this one: Do 60% of US anti-vaxxers identify as politically liberal?.
My answer here concerns anti-vaccination opinions generally, not only the specific phenomenon of belief that vaccines cause autism. My impression is that autism fears are a smaller fraction of anti-vaccination opinions now than they were when this question was originally asked.
Probably not, but it may depend on exactly what sort of anti-vaccine opinion you look at.
Here are the relevant sources I've been able to find.
Here's some information about a Pew Research survey conducted in mid-2016. So far as I can tell, the actual data aren't available. (If anyone else wants to look, this seems to be the May 2016 "wave" of the "American Trends Panel".) But the report says:
Republicans (including independents who lean Republican) hold roughly the same views as Democrats (including leaning Democrats) about the benefits and risks of the MMR vaccine.
However, political conservatives are slightly more likely than either moderates or liberals to say that parents should be able to decide not to have their children vaccinated.
The figure seems to be 25% for conservatives, 16% for moderates and 9% for liberals.
There's also an audio interview with someone from Pew which I think is about the same survey; it doesn't give any more information about correlations with politics.
So this shows little difference between conservatives and liberals, but what there is suggesting conservatives are more opposed to compulsory vaccination.
Here's some information about a Pew Research Survey conducted in August 2014. This was of young adults only; it shows 34% of Republicans and independents but only 22% of Democrats saying that parents should be able to choose whether their children should be vaccinated. (Larger figures than from the survey above; I guess because of the different population surveyed. The interview above mentions that younger people and parents of young children are more often dubious about the benefits of vaccines.)
So this shows *little difference between conservatives and liberals, but what there is suggesting conservatives are more opposed to compulsory vaccination.
Here's some information about a Pew Research survey conducted in February 2015, specifically about views on the safety of the MMR vaccine. 5% of Republicans, 9% of Democrats and 10% of independents said they didn't think the vaccine was safe; the survey had n=1003 so the differences could plausibly be chance; also, there were more "don't knows" among the Republicans. (It doesn't seem like autism worries were a major concern for most of these people, by the way.)
So this shows little difference between conservatives and liberals, but what there is suggesting liberals are more skeptical about the safety of vaccination.
Here's a paper about a Mechanical Turk survey of 367 Americans (important note: Mechanical Turk workers are far from a random sample of the population) asking about politics and vaccines. The researchers found that respondents identifying as liberal were significantly more likely to agree with pro-vaccination statements and to say they'd had their own children fully vaccinated.
So this shows liberals being less anti-vaccination but in a possibly-unrepresentative sample of the population.
Pew Analysis (2017)
Here's an article by someone who's analysed some of the Pew data and says what it shows is that more politically polarized people on either side are more likely to be anti-vax (to about the same extent on each side) but that conservatives are more likely to favour making vaccination optional and liberals aren't. (The author suggests that the latter is because conservatives are more likely to distrust the government generally.)
So this shows little difference between conservatives and liberals, but conservatives being more opposed to compulsory vaccination.
Here's an article based on a survey apparently conducted by CivicScience — unfortunately I have not been able to find any further information about this survey. It says that about 7% of the ~2300 respondents called themselves anti-vaccine and that 60% of those people describe their political leanings as liberal.
This does appear to show liberals being more anti-vaccination; it looks like it's focusing on a narrower sort of anti-vaccination-ism than most of the other references above.
Paper: "The Role of Conspiracist Ideation and Worldviews in Predicting Rejection of Science" (2013)
Here's a paper about a range of "anti-science" views which finds that anti-vaccine opinions are positively related with "free-market ideology" but negatively related with "conservatism". The authors suggest that what's going on is, again, that libertarian views go along with wanting to let parents choose.
This may kinda-sorta show liberals being more anti-vaccination together with (one kind of) conservatives being more inclined to oppose mandatory vaccination.
Paper: "Vaccine Risk Perceptions and Ad Hoc Risk Communication: An Empirical Assessment" (2014)
Here's a paper about anti-vax positions from 2014; it finds that "respondents formed more negative assessments of the risk and benefits of childhood vaccines as they became more conservative and identified more strongly with the Republican Party" but that this effect was small. (It also finds that anti-vaccine sentiment is importantly different from other sorts of "anti-science" sentiment and argues that it shouldn't be treated as "anti-science".)
So this shows little difference between conservatives and liberals, but what there is suggesting conservatives are more skeptical about the safety of vaccination.
Here's the data from a PPP poll in March 2013 — finally, some actual raw data!. Subjects were asked whether they think vaccines cause autism. When classified by political leaning, the results were nicely symmetrical except that nearly twice as many "very conservative" as "very liberal" respondents said that vaccines cause autism.
So this shows little difference among political moderates, but more skepticism about the safety of vaccination among the very conservative than among the very liberal.
Putting all the above together, the picture that emerges (but remains extremely cloudy) seems to be something like this.
- Moderate skepticism about vaccination seems to be pretty much non-partisan, with perhaps a bit of a conservative lean.
- More extreme anti-vax sentiment — the sort that is found among 5-10% of the population — seems to lean liberal, but the evidence for this is poor (small n, surveys with no information to speak of actually released).
- Where research distinguishes between (1) concern about vaccine safety and (2) opposition to mandatory vaccination, #1 seems to lean left and #2 seems to lean right. (And in the one case where the research distinguished between "conservative" and "libertarian" among rightish views, it was the latter rather than the former that seemed relevant.)
- Most anti-vax opinion these days doesn't seem to be about autism.