Can I allow customers to purchase out-of-stock items on backorder on Shopify?

Shopify doesn't have a built-in feature that allows you to create backorders for products that are sold out, however, you can make use of the "Notify me when available" button, which will notify your customers when the desired product is back in stock.

Alternatively, you can make use of one of many Shopify Apps that specialize in backorders. Some of these apps you may want to take a look at include Back in Stock and Pre-Order Manager.

You should also make sure that your sold-out items are not hidden, as well as add a sold-out badge to your best sellers and provide customers with alternatives to sold-out products.