The vaccine is free, even without insurance. Anyone can walk into a CVS and get it. I don't see what the health care system has to do with that. That's actually an instance where privatized health care should work better than a government controlled one - and it does compared to a lot of other countries. The US does much better than many European countries for example.
I don't think whether socialized vs private aspect of healthcare has much effect on vaccination rate one way or another. The highest vaccinated among big countries were for a long time US and UK, one with private healthcare and the other with a single-payer system. UK was actually ahead of US for most of the time. Current vaccination rates in European countries are within a few percentage points of each other, whether they have a private system (e.g. Switzerland and Germany) or socialized (e.g. France).
