If you just want the user to be able to designate invoices as "paid", you can add a text field to your invoice table and give it a checkbox set control style. You can specify a value list with one custom value: "Paid" for this check box set.
If you want the database to automatically mark invoices as paid as payments from the customer are logged, that can also be done, but the details can get a bit tricky depending on how you permit your customers to make payments. Can one payment pay off several invoices? Can a customer make partial payments to don't completely pay off the invoice?
Working off Phil's comment, on the invoice record you could put an AMOUNT-PAID field. That subtracts from the total owed field of the invoice to give a difference. Using a calculation field you could have a case statement checking conditions: Paid balance still owing, Paid in Full, Over Paid.
If you have multiple occasions when many payments are used to pay a balance off, you could have a payment table that is summarized in the AMOUNT-PAID field referenced above, changing it from a number field to a calculation field that sums related amounts from the payment table.
Thanks for both of your suggestions. I will work on it this weekend. Appreciate the help.