When the Empire left Argonia, the large plantations almost surely departed with it. These Imperial plantations are the only reference to any sort of organized economy that I could find in the province, but what of the native economy?
I tend to believe that the Argonians are a society of primitive hunter-gatherers, since the environment apparently prevents any sort of agriculture from functioning efficiently. In addition, there likely aren't any goods worthy of exporting, except perhaps alchemical ingredients.

What do you picture the Argonian economy leaning on?

The http://imperial-library.info/content/argonian-account covers a lot. Fruits, berries, probably fish and crocodile, maybe insects for the Dunmer and weird Nibenese. Coal? There's probably plenty in Argonia (assuming it forms as in our world), but I haven't seen any indication of it's existence in TES.
I'm guessing that the plantations were gone before the empire was forced out left.