If you see an unwieldy person coming down the aisle of the bus with an unwieldy box, you may want to step aside because that's a double dose of clumsy. It's an awkward person carrying a box that is difficult to manage.
Something that is wieldy is easy to control or handle, so something unwieldy is not. However, the un- form of the word is much more common. The base word wieldy has its roots in Old English, meaning "to handle or control" — used usually in reference to a weapon. You would not want an unwieldy person attempting to wield an unwieldy sword!