> Let's say it was critical for the business, with no viable alternatives?

If its that critical for you why are you rolling the dice on a general email and ... waiting for an email back and forth? Something so important as a vender "license" deciding if your company succeeds or fails should not rely on an email sent to an address pulled on their website.

Find a contact, a human, drive (or fly) to their office and make an in-person appointment, show they are important to you. Otherwise that game of AI email tag should be more than enough to tell both parties just how not serious the whole thing is.