I'd like a tag to be applied to an order as soon as a certain customer places the order.
Yes! There is an option in the trigger conditions that will allow you to have a tag automatically added to an order as the order is created.
You can do this by creating a new trigger (Settings -> Triggers). Then you'll see you can have this trigger fire for al companies in your system, a group of companies (via tags) or a single company.
Then in trigger conditions you'll choose the order status option for when you want it to fire (in this case 'ordered' status).
And finally, you'll choose 'add tags' in the action section of the trigger and put in the tags you want to be added to the order.
In the image below you'll see this particular trigger will fire when any company place an order (or moves a bid up to ordered status), and it will add two tags to the order ('ashley' and 'report').