If you've got proof that you complied with their system the first time, i.e. email confirmation, or a screen print of the completion confirmation, or similar, then tell them to swivel and start debt collection procedure.
Otherwise, you're on shakier ground as you've no proof you did it.
These days, with more an more automated accounting, lots of firms, even smaller ones, are using online systems requiring invoices to be uploaded to portals etc along with "proof" the work has been done, etc etc. A couple of years ago, they were pretty rare and only for bigger organisations, but now most businesses, even the smallest, are using online semi automated accounting, this is becoming pretty normal.
You could easily end up with legal disputes because whilst the customer didn't include in their order acceptance, their T&Cs saying the invoice processing procedure (I presume they didnt), I presume your own T&Cs don't say that you'll expect payment x days after sending them a paper invoice either (I presume they don't). So there's ambiguity as to payment procedure from both sides, I'd expect!