How can I create a company-branded “debit card” that works at standard point-of-sale terminals?

You can just have a singular "card association". To have a debit card for your own company, you just need to have the numbers. You can go to a pre-paid debit card provider or to an issuing bank. Both will provide you with a card associated card, provided your business case makes sense to them and you have the minimum qualifying numbers for card commits in their territory.

Nothing really to it. A VISA/MasterCard debit card will work pretty much on all networks.