Magento 1.9- How to allow orders for out-of-stock products?

Magento has in-built setting for turning on "Backordering".

By changing the settings in System > Config > Inventory will only affect the defaults when you create new products, you still need to enable each product to allow backorders.

Easiest way to do this is select all products you want to allow to be backordered, then select Update attributes from the actions drop down and click submit.

Once the screen has loaded select the Inventory tab from the top left. Then select the change box for 'Backorders' and then click save.