Both the Retailer and the Supplier can process a refund for the customer. Canal recommends discussing with your Retailers how you prefer these situations to be handled.
Retailer processing refunds
If the Retailer is hosted on Shopify, refunds will flow from their Retailer to the customer. They must separately request a refund from you. If their Retailer is using Canal's API, then they will create a refund request using their refunds endpoint.
Supplier processing refunds
If you will be processing the refund, you will issue this as you would for an order that came through your site. The refund will flow from you, to the Retailer, to the customer.