I believe I may have already emailed you about this subject, but while we do not have a delivery date feature in our cart it is possible to offer multiple shipping options in the cart so that a buyer has a choice between "express" or "regular" shipping... or "same day" "next day" etc.
What the shipping feature really does is apply a cost that you define to the order, either a regular baseline cost for all orders or a special cost for expedited delivery, etc. That would be the core of what you need in a Delivery Date tool anyway, because our cart doesn't actually do any of the shipping for you or integrate with any shipping services directly. All a Delivery Date tool could do is tell the buyer when to expect their shipment and apply an appropriate cost to the order for that level of service.
You can read more about our shipping features here, and if you can give us a detailed description of what shipping costs need to be applied in what circumstances we can give you instructions on setting that up: