Is there a way to have the cards arrange themselves automatically based on the due date that has been set?
No, there is not. Currently there is an auto-arrange by vote idea that is very highly rated, and it might be good to add this request to that conversation, as the two would be very similar: https://trello.com/card/board/way-to-sort-column-by-votes/4d5ea62fd76aa1136000000c/4dd28b64728b921a5d1e7209
You can't auto-arrange cards by due date, but, there is now a card view to display cards by due date.
To set it to sort fully automatically within Trello is currently not possible, no.
With the Chrome extension Ultimello however, you may do that manually. It's still very valuable for large columns of items.