Probably they wouldn't as full audits aren't that common, but audits do happen and most accountants won't care about your purchase as much to personally commit a felony and forge documents for no good reason (accountants do have personal responsibility, and every accounting course reminds wannabe accountants that "the boss ordered it" is not an excuse) so they simply say that they can't/won't do it and if the vendor can't send a satisfactory invoice then they won't do business with that vendor.

How would a French audit uncover anything wrong with a conforming format invoice under the name of some random American company that exactly matched the bank statement, physical goods, and customs paperwork?

Hey notch656a! I'm not a government expert, but the challenges would start with:

- I am Frenchy (a French co) - Buying software from Ricky (a US co)

If I don't have a proper invoice from Ricky to give to my accountant at the end of the year, and I get a tax control, then, I'll be fined.

If the tax ministry notices large amounts of similar fines, they might look at Ricky.

So the risk is more on Frenchy (risk to be fined). But I'd say that Ricky is safe for a while, unless the invoiced amount are really huge.