Processing a payment in this case is a form of updating a record. Which basically means it changes the invoice status to "paid". When you access the form directly you are only able to create new records (which is not what you want). Instead, you need to add the editing formula to your Airtable. It would generate a unique URL for each record and you can then share those URLs with your clients. That would allow them to click on a link to open the invoice and pay it.
Make sure to watch all these 3 instructional videos in order:
They guide you step-by-step on how to set this up.