Hi there,
You can setup a new order notification to be emailed to any email address (or multiple addresses) you want but, by default, somebody will still have to login to the admin and mark that order as complete. If this is an issue, you can also have a customization done to include a link into admin order notification emails, something like "Click to complete". Clicking on that link will complete an order without login to admin. So an order can be marked as complete with one click even from a mobile phone. It will require some custom work but possible.
Another option is to connect Magento to your ERP system (if you have one) so online orders will go into that system and will be managed there the same way you manage phone or walk-in orders.
You can also connect Magento to many POS systems and it may be possible to manage online orders from your POS (depends on the system)
Hope this helps.