Depending on what e-commerce platform you use and what supplier you want to use, there are usually app integration which you can install on your store to automate your order Fulfillment process, however if your supplier doesn't have any app/plugin integration, most platforms (i.e Shopify, Woocommerce, Ecwid etc) allows you to add the supplier as a staff on your store with order permission, so that whenever you get an order, the Supplier gets an automatic order email notification and can fulfill it.