Can customers ask to be told when a product is back in stock?

Yes. Enable the In-Stock Notifier plugin. On an out-of-stock product page, shoppers leave their email.

• When you restock through Inventory (warehouse or store stock), they are emailed automatically.
• You can also send the email yourself with Notify on the plugin's subscriber list, for example after updating a product's quantity directly.

The list also shows you which out-of-stock items people are waiting for.