In their company settings you can add credit card and bank accounts (the same place you go to change their address or their permissions, etc). You'll add the bank or credit card payment methods by using the "Manage Payment Methods" link in the settings:
Adding a bank account will allow them to use the "quick check" option when they make payments. You don't have to set anything up for a cash option or regular check option - in that case they'll give you the cash or check and your staff will be able to process the payment with the "Offline Payment" option. This allows you employees to choose whether they are paying with cash or check, etc.
Customers can also add & edit their own payment methods too.
When they log into their Allmoxy account, click on the Settings icon and then select payment methods. Then they'll be taken to a page where they can add a new one, or delete an existing payment method if they need to update or change out the information!