There are two scenarios here to manage the same.
1. Complete rejection of Order:
>>If any item of the orders is not available, your entire order would be rejected by Amazon in our app.
So, you can ship/ fulfil from your own warehouse in Shopify manually this order.
2. Separate Partial Shipment/ Fulfillment:
>>If any item of the orders is not available, your entire order would be rejected by Amazon in our app.
So, you can fulfill one item which is out of stock from Shopify manually and then with order items (available in the Amazon warehouse) you can re-submit this order in our app and this would be synced only with Amazon items/ SKUs successfully.