I’m Helen from Shopify Support. Thanks for your question.
There is an app in our App Store that includes the delivery date on the customer order confirmation email; It is called On Time Delivery. This app is created by a third party developer, and is fully customizable. There is a free 3-day trial for this app, therefore I encourage you to add it to your store and test it out to be sure it functions as needed. Uninstalling the app before the trial ends, will ensure that you are not charged for it. Any questions about the functionality or set up are best answered by the developers, so please reach out to them directly if needed.
Alternatively, we have a Help Doc to add a delivery date picker to your cart that walks you through the steps to add a delivery picker to your theme code. It is possible to then customize your order notification email to include the chosen delivery date. If you are using a Shopify theme, are on a paid plan, and have design time remaining on your account, our Theme team can help to customize your order confirmation email to include the date your customers choose by using this date picker coded into your cart page. For more information on getting assistance from the Theme team, please see here.
If you are not eligible to use our Theme department for these customizations, I suggest reaching out to a Shopify Expert. You can find one from your Admin where you will submit a job request and discuss your requirements with any Experts that are able to help you. More information about finding and hiring experts can be found in our help center.
All the best,