By default the address book is not able to be changed at the device, you can somewhat enable it, but it will then only allow you to add an entry, or edit a current one, but not remove any.
You have to enable it for All Users here in EWS
But, as stated, you will only gain add and edit as functions
Is it possible to delete a contact from the device address book using the device's touchscreen? If the only method is from the web interface, does the user have to be logged in as the admin user or can an account be setup to just be able to edit/manage the device address book?