How to determine the source of PVA: crude or plant-based?
It certainly would be possible to get an indication of some of the sources of carbon through mass spectrometry and through radiocarbon dating. For example, fossil carbon sources would have little 14C, sources predating the 20th century, e.g. older trees, would have more. Post-Atomic Age sources might have more yet, beyond that due to natural decay of 14C.

That said, it's a nice question, but are you prepared to pay for the analysis to have it answered? Perhaps your government might perform such testing to verify a claim of being "organically produced".