Comparing believers of Young Earth Creationism and anti-vaxxers is a false equivalence.
Creation Ministries, especially, has always been careful to distinguish operational science and historical science. The study of immunology fall under the latter; the experiments involved in that field are repeatable and reproducible, and therefore the reports on the efficacy of a newly-developed vaccine is easily falsifiable by a different laboratory/research team testing the sample of the vaccine on a sample of the virus strain and see whether or not the product works as advertised (although, of course, in vitro and in vivo experiments may produce different results due to the variety of the human body response in different individuals -- which are in turn affected by a myriad of other factors from genetics to pre-existing health conditions -- and this adds the complexity of verifying any scientific claims, especially when it involves health & wellness).
The evolution vs. creation debate, meanwhile, is a historical science. It is neither repeatable, reproducible nor falsifiable, as we cannot re-create the Big Bang or whatever origin story one believes in, and as a result the proponents of any such theory would rely more assumptions and presuppositions (i.e. faith) in order to reach a conclusion on the narrative. History, after all, are based more on eyewitness accounts, testimonies, written records such as letters and memoirs, rather than actual empirical evidence. And while the origin narrative a person holds can have philosophical implications that might influence the rest of their worldview, it doesn't determine the authenticity of their scientific authority -- just as a programmer doesn't need to know how their computer and the history behind its invention, to be able to create a functioning software.
As for the issue of fostering science distrust, I think it has more to do with the people's general wariness against government bodies, large institutions and media outlets who tend to have something to gain by promoting their version of truth claim (whether or not it actually true) and have the power to get away with the consequences if they are disproven. What they believe about the origins of the universe seems hardly a factor in these issues.