Whenever I've paid the co-pay at the doctor/dentist they have always written a receipt. Both our doc/dentist use the kind where a carbon writes a line on the ledger sheet. I would imagine that all offices would have to have some type of receipt/accounting. They should be able to easily see if the patient paid the co-pay or if being a newbie you simply forgot to ask for it. That has also happened to me.....and I usually get a bill in the mail for the co-pay because either they forgot to ask for it and I forgot to offer it....or once I didn't have any cash or my check book. The doc charges a service charge for paying via credit, so they agreed to bill me instead.
I totally understand your nervousness being your first week.....but keep your chin up, you know you didn't steal it, so there is a simple explanation for it and then you can apologize for the error in not collecting it, offer to send a bill to the customer or call them to request they send payment.