Both the Storefront and the Supplier can process a refund for the customer. Canal recommends discussing with your Storefronts how you prefer these situations to be handled.
Storefront processing refunds
If the Storefront is hosted on Shopify, refunds will flow from their Storefront to the customer. They must separately request a refund from you. If their Storefront 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 they would for an order that came through your site. The refund will flow from you, to the Storefront, to the customer.