What if Customers are able to checkout without selecting the delivery date – when checking out from the cart drawer/popup cart?

Order Delivery Date App does not work with Ajaxify (cart drawer). It only works on the full Cart Page.
Using a Drawer/Popup cart type of theme?
If you want to change the Drawer/Popup cart into a full cart page:
Go to Themes > Customize Theme and set cart type to Page (or ask your theme developer how to disable the popup cart). App work only on the full cart page.
If you still want to use Cart Drawer / Popup cart, we support the third part app iCart Cart Drawer Cart Upsell, with this app, we allow Date and time picker inside the cart.