If your supplier's feed uses an item ID that doesn't match the SKU or Barcode field in your Shopify store, you could use a Variant Metafield for the mapping instead.

In EZ Inventory, when creating a Data Feed configuration, simply select the option "Variant Metafield" under the Field Mapping > Shopify Variant Field option.

You'll be provided with an extra option to specify the "Metafield Namespace and Key".  Just enter the Shopify metafield's namespace and key separated by a period here.

You can find the "Metafield Namespace and Key" in the Metafield Definition in the Shopify Admin. 

For example:

Please note that this could significantly slow down the processing if you have a lot of variants in your store as pulling metafields data requires additional API requests.