You are referencing his own web site/his claims, but as he seems to be the only one making his claim of invention, that does appear to be the only source for the sweeping claim itself.

But I was able to deduce the source of his claim from the highwire video. It is as I suspected. He is one of a fair number of people that worked in this area (he did some of the early research on mice) and is now trying to claim the lion share of the credit.

This very loose type of credit-claiming can be debated endlessly, but it appears most of his work in this area is confined to the late 80s and early 90s. Yet actual mRNA vaccines weren't created till much later. Inventor implies "reducing to practice" and he certainly did not do that.