If all those machines were on the same network, then yeah, I imagine it has something to do with that. Don't know what it would be, though.
I can try asking our ecommerce provider but I don't think they'll be able to tell us why it's not working, either.
Do you have a PayPal account? If yes, could you try using that instead? If it's getting stuck on the credit card authorization for some reason (which is what happens when you click the "next" button on that page), then using PayPal instead would bypass it. From what I can see, your orders aren't making it to the authorization stage.
Worst case scenario, if it's still happening tomorrow, I can try to help you complete the order.