Product stock status in WooCommerce is stored in the _stock_status private custom field, which can have two values: instock and outofstock
On the search instance Frontend Search Settings -> Custom field filters panel you can easily make a filter on this field:
..this will result in:
To the Custom Field input enter: _stock_status
In this example I use the Radio type, but Checkboxes or Drop-down is just as good
To the values textarea enter:
||Any value**instock||In stockoutofstock||Out of stock
For more information about custom field filters please visit the Custom Field Filters documentation.