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