Will customer email be sent if a product is set to "Allow purchases when not in stock"?
Back in Stock will send the customer email if the variant has an inventory quantity greater than zero. If you've got "Allow users to purchase, even when sold out" set the inventory quantity - could be set to zero (or lower) and Back in Stock will not send email.
To send customer notifications for products set to "Allow users to purchase, even when sold out":
- From your Back in Stock dashboard choose Account
- Find the option Send notifications for products that are sold out but available to purchase and make sure it is checked.
- Hit Save
Enabling this setting will also allow customer notifications to be sent for products that do not track inventory.