Can the Element API export Craft Commerce orders as the element?
Orders are in fact an ElementType in Craft.

See Commerce_OrderElementType.php inside the commerce plug-in folder. With that said you should have no problem retrieving orders like you would any other element type in Craft.

There are also services available that make it very easy to retrieve orders so you may find a simple PHP plug-in will get the job done in terms of returning JSON for an order(s).