First, are Reps more opposed? Apparently so - this study doesn't cite the numbers per party clearly, but you can see the state by state breakdown strongly correlates.
To understand why, you first need to compare covid to the 1918 Flu epidemic, rather than to other vaccinated diseases. During that Spanish Flu, there was some differentiation between take-it-seriously (and hurt the economy) regions/cities and keep-the-economy going areas
Privileging the economy and companies is usually a Republican differentiator. So is opposing government oversight that is believed to be unnecessary.
And here's some more Pew Research goodies, dated from July.
Majority of Republicans say country has given too little priority to respecting individuals’ choices during the coronavirus outbreak
For instance, 75% of Democrats say COVID-19 vaccines have been extremely or very effective at limiting the spread of the coronavirus; 16% say they have been somewhat effective and just 9% describe them as not too or not at all effective.
Republicans offer a much more skeptical view: A slightly larger share of Republicans say vaccines have been not too or not at all effective at limiting the spread of the coronavirus than say they have been extremely or very effective (39% vs. 32%); 29% fall between these two views and say vaccines have been somewhat effective.
Trump made an early call to shut down travel with China and later Europe, along with a circuit-breaker lockdown. This was in hindsight a good call, but it quickly turned into a political liability, with among other things a Lt Governor in Texas stating that old people should not be afraid to lay down their life for the country and take risks.
By Easter Trump wanted to open back up and many measures to slow down covid were deemed "too Democrat" by Reps. Witness the mask and distancing drama in places like Florida. The fact that the brunt of the early deaths were in Dem New England also took away much urgency. So - government control on covid => bad (in most Republican cases). Trump follows that narrative from May on, downplaying the risks of the disease and staying well in line with his base. He certainly didn't want to do anything that would put his reelection at risk.
Trump starts vaccine research right away - again a good call. But it doesn't, can't, arrive in time to save him at the polls.
After the election, any attempts by the government to push vaccines is - you guessed, Dem governmental overreach. The examples of Europe (Macron) and Canada to push passports were not followed, but there were some attempts at mandating within industries, thus reinforcing negative perception of vaccines.
It would have helped if Trump had stood up for "his vaccines" in late 2020, early 2021, but IIRC he even initially demurred even to state if he had been vacced. By then, Trump was probably hesitant to appear "weak"
** to his followers. Vaccines could have been adopted as a low-cost, low-risk, way to get back to pre-Covid normalcy, but that message was never pushed by most Republican politicians or pundits, only the narrative of government overreach.
It's not so much that Trump himself was anti-vaxx - he certainly talked them up from May to October, saying that would come in any time and save the day. It is that, past his electoral loss, he lacked the leadership to use his influence to promote his own administration's vaccines to the Republican electorate, in a non-mandated way. He could have said "No mandates! But, if you want, roll up your sleeves to bring America back to normal, save businesses! I made them, they're the best vaccines!".
I will also add that, unlike 2015 and before, the conspiracy/science-denial belief has swung firmly into the Rep camp. While individual anti-capitalist subgroups hold conspirational beliefs against Big Pharma and the like there is nothing to compare in breadth and looniness to QAnon's beliefs systems, which are influential enough to have several candidate lawmakers campaigning on their tickets. Few Reps will directly believe in QAnon - shape shifting aliens and Satanist pedophiles are a bit much - but they will have many more opportunities to be exposed to its corrosive messages than the average Dem will have from lefty anti-vaccine conspiracy groups.
p.s. This was answered from the US PoV. Canada has more or less the same divide (our right absorbs quite a lot of political thought from the US). No idea how other Western countries figure, but I doubt it's as clearcut.
p.p.s It is not necessary to give me all sorts of feedback about the efficacy, or not, of Trump's early covid measures. The point here really isn't whether they were efficient or not - the point is that they were considered too restrictive on businesses and personal liberties by a significant proportion of Republicans (see my 3rd paragraph from top as to why).
* The Great Influenza IIRC pretty much went on the record that strong anti-flu measures correlated with less long term economic disruption in 1918. But it clearly stated that it had been a subject of disagreement.
** Esper's Sacred Oath goes on at length how Trump hated to give the appearance of personal or national weakness. There are documented cases, such as the initial response to the Charlottesville car attack, where Trump initially said something more or less centrist, before tacking back to the right when called out on it by Fox News and the like.