From what I see you can set up AyMakan to send webhooks with a tracking number. This makes it possible to create an integration with Integromat or Zapier. First step is to receive the tracking number through a webhook, and then create a fulfillment in your Shopify store. Shopify will then send a shipping confirmation email to the customer.
How do you create shipping with the AyMakan API in the first place? It'd be crucial to fill in the reference field with order ID so that you can then create a fulfillment for an appropriate order.