Are you sure that you are properly authenticated with the API? I assume you have entered your credentials in the Settings page of the Constant Contact options, but maybe it isn't working.
When you look in the Constant Contact section of the sidebar, do you see all the extra pages like "activities" "import" "export" and "lists"? If you aren't logged in properly that might be why the widget doesn't display.
Also just in case: did you try adding a different widget to the same sidebar to see if it displays? It could be your sidebars aren't configured like you think they are.