If your supplier uses a completely different SKU or Barcode from what you use in Shopify, you could potentially use Variant Metafields to create a custom field in Shopify where you can store the SKU/Barcode for this specific supplier.

Once you've added the Variant Metafield in Shopify, you can then configure EZ Inventory to map to this custom field.  Please click here to learn more on how to configure this.