I'd be interested in the answer to Scott's question too.
I have a prospect working with teamwork. What is the best way to connect to this application?
Does Qlik's RESTful API actually work to retrieve data from teamwork?
Alternatively, is there a premium connector from Qlik or a reliable 3rd party?
If you didn't get it working yet. And for other people searching:
You can do a normal REST connection GET call to the url of your teamworkpage and add /projects.json for example. to get the Project data.
use basic authentication with your API key as the Username and no password. skip certificate. And you will be done. check the teamwork API website for other calls. It is really easy.