The Coaches Console does not process payments and does not store card data. It is only a go-through environment for payments. If you want your clients to be able to pay your invoices via the Console and have the invoices marked automatically as paid in their Invoices section, then a merchant account with Authorize.net or/and PayPal is needed.
So, the system is intended for use with a merchant account for payments. However, it is not necessarily a requirement if you don't mind the clients paying the invoices outside the Coaches Console. They could pay you with checks, cash, bank or wire transfers, on other payment platforms (Stripe, etc.), you can waive the fees or do bartering and/or pro-bono client scenarios.
However, in all these cases where the clients pay your invoices outside the Console, the invoices will not be marked automatically as paid in the clients' Invoices section, so in order to reflect the payments in the record, you will need to mark them manually as paid (in CONTACTS TAB > Contact Name > Invoices tab > VIEW option in the drop-down menu to the right of the invoice > button "Add payment" in the invoice pop-up > setup the details > button "Apply").