The answer (and implications) depends on who you ask. There are those who reach for the limited available statistics, like @Bobson's fine answer above. I've got no objection to the approach, but the limitations are in the pollster's intent.
On the other hand, while I'm fully vaccinated, I've discussed the matter with friends who have chosen not to accept the vaccination(s) and had some success in influencing them to get vaccinated. Based on (admittedly anecdotal evidence) I think the answer is yes, there is correlation.
Specifically, it seems that opposition to vaccination can be correlated with distrust of government. I would even go so far as to say that distrust of government causes opposition to vaccination. This applies specifically to the COVID-19 vaccines. I've got no statistics to back that up at this time, but I'll dig into it a bit.
The Biden administration has taken a very different approach through the White House press secretary. When asked about the president's poll numbers and COVID, she seemed to blame those who don't accept the vaccine for the President's dropping rating. While this seems to be similar to my conclusion on correlation, the causality implied is reversed from what my experience indicates.
I suggest that both distrust in vaccines and low government job approval are rather obviously correlated and caused by the level of distrust in government. In addition, distrust of big pharma is another major factor, so information from the government regulatory agencies (who work closely with big pharma) is a primary factor in the crisis of confidence.
There are lots of good polls with data showing a correlation between trust in government and other political beliefs. Perhaps it's obvious, but there is a general trend for those aligned with the party of the current president to trust the government more than those aligned with the opposition party. I recommend the Pew Research Center's Key findings about Americans’ declining trust in government and each other and more specifically Public Trust in Government: 1958-2021
The other factor to consider is that there's a substantial distinction between people who are opposed to vaccination, and those who are skeptical of the government's approach to managing COVID-19. Take for example this discussion between prominent doctors & scientists, all of whom believe in (and have been) vaccinated, but who raise serious questions about the government's strategy of focusing on vaccination.