This is not my understanding. PCR works to produce DNA because the DNA polymerase enzyme creates DNA copies from DNA templates, and those copies become templates themselves, feeding back into the chain reaction.
I'm not aware of similar enzymes capable of making RNA copies from RNA templates (RNA polymerases use DNA templates AFAIK).
I was imagining the RNA vaccines would have used a fully synthetic oligo production method, to get the extreme purity required. But it might be that some sort of PCR-like amplification process is used. Would love to read more if someone has details.