The only thing you can try is to enable this setting in "Settings"->"Checkout", if that doesn't work then yes you would need Shopify Plus to add custom fields and include custom scripts and events.
There's another problem:
I enabled the field for additional information. It worked when I open the checkout page.
Then I placed a test order with additional information (Resident ID Number).
I opened the order detail page in the orders admin. Additional information was on the right side of page.
However when I refresh the page, additional information was gone.
The checkbox for additional information also got unchecked automatically.
Step 1: Enable fields for additional information
Step 2: Fill out checkout information
Step 3: Place order
Step 4: Order was placed
Step 5: Open orders admin page
Step 6: Open the order detail page (Additional Information was on the right side)
Step 7: Refresh the order detail page (Additional Information was gone)
Step 8: Checkbox got unchecked automatically