Hello,
You could use our Variations feature for this, that would allow you to create a product button with an attached text field, your clients could enter their invoice number there along with the amount and submit the order online. Whatever they enter goes into your sales records and is included in the notification we send you.
Keep in mind that we have no way to verify the invoice number or the amount they are entering, so you're going to have to trust them to get that right. This also goes through our cart as an order so you might want to make sure that the information we send them by email afterward doesn't use the invoice number we'd generate for the transaction, since that could cause some confusion.