I'm more of a music history expert than a coffee expert, but I can at least get you started on some information that might help.

A quick read-through of this paper tells us that in the early 19th century, most coffee that found its way to Europe had been imported from European colonies (particularly those of Spain). European colonialism apparently helped to somewhat industrialize the process of importing coffee, along with modern advances in transportation. Originally, Austria (where Beethoven spent most of his life) was first introduced to coffee from the Ottoman empire, but it seems that by Beethoven's day the process of importing from European colonies reigned supreme.

This page tells us that during the time Beethoven was in his young adulthood, Latin America (and Haiti, in particular) were the biggest juggernauts in terms of coffee production and export. So, it seems that it is very likely that Beethoven drank coffee made from beans imported from Latin America; and quite probably, from Haiti.