The oral polio vaccine is administered without the use of needles, and therefore could serve as a testbed for this hypothesis. Unfortunately, I didn’t find much literature addressing the question directly of how much more people are willing to take an oral vaccine compared to a needle-based one.
There might not be a clean RCT for this but just looking at the history of the Polio vaccine, I seem to find confirmation for this. In the West, the Salk vaccine (which had to be injected) was available since the 1950s but uptake was very slow. Then the Sabin vaccine was developed around 1960, mostly in the USSR with the initial idea from Sabin (a naturalized US citizen), which was an oral vaccine and a much bigger success. This is confounded by the fact that the Sabin vaccine was considered more effective, but on the other hand it also had a slight chance of giving others a real infection because it contained live viruses. So, I don’t know how that affected its perceived desirability. Still, I’d consider this slight evidence in favor of oral vaccines being more popular.
There might not be a clean RCT for this but just looking at the history of the Polio vaccine, I seem to find confirmation for this. In the West, the Salk vaccine (which had to be injected) was available since the 1950s but uptake was very slow. Then the Sabin vaccine was developed around 1960, mostly in the USSR with the initial idea from Sabin (a naturalized US citizen), which was an oral vaccine and a much bigger success. This is confounded by the fact that the Sabin vaccine was considered more effective, but on the other hand it also had a slight chance of giving others a real infection because it contained live viruses. So, I don’t know how that affected its perceived desirability. Still, I’d consider this slight evidence in favor of oral vaccines being more popular.