This seems like a non-partisan thing that only became partisan recently. Look at RFK Jr, a lifelong Democrat who pushes anti-vaccine misinformation.

Here's how I see events:

1. GOP has White House when COVID happens and they push for vaccine availability
2. Democrats criticize GOP response saying vaccines should be mandatory
3. GOP pushes back saying it should be optional
4. Anti-vaccine groups latch on to the GOP since they both oppose mandatory vaccination

If Democrats were in power, they likely wouldn't have pushed for mandatory vaccination, so the anti-vaccine crowd probably wouldn't have allied with the GOP (I'm guessing the GOP wouldn't criticized slow development of the vaccine or something).

Most people are vaccinated, so vaccination is absolutely the majority in both parties. Anti-vaccine groups are like any other conspiracy group, they'll join with whichever larger group seems to be more sympathetic at the time.