Yep. And to a somewhat lesser extent, so does international commerce and logistics. Nearly all major importers of goods (think Wal-Mart for example) and their various suppliers and service providers rely on EDI to exchange commercial documents (purchase orders, commercial invoices, packing lists), file customs entries, track their incoming shipments and inventory, etc etc.
EDI is probably irrelevant and dead in Silicon Valley (unless maybe you work at Oracle). But in healthcare and commerce it's very much alive.
It exists heavily in my industry (retail channel management) but mostly as design specs. Most companies tend to use XML it seems but they still base their feeds off of the old-school spec.
For the record the term is still thrown around left and right in enterprise settings.