You most likely need to add "127.0.0.1" to the list of allowed domains on your kit in Typekit.
See these articles for more information:
(Scroll down to "How to Embed Web Fonts Using Typekit")
Don't forget to publish your kit after adding 127.0.0.1. Also, note that it may take a few minutes (I noted up to about 15 minutes) before the changes to your kit will be completely published.
Thank you, Carey adding "127.0.0.1" to the list of allowed domains on your kit in Typekit made all the difference and the whole article http://blogs.adobe.com/edge/2012/01/27/web-fonts-and-edge-together-at- last/ was really helpful too.
i added 127.0.0.1 to the list of domains in my kit & republished a bunch of times yesterday, and they're still not showing in my edge animate file.
I think I may have the same issue. I have added localhost / 127.0.0.1. The fonts appear fine when previewed in the browser but does not show correctly inside the app.
turns out i had the font named incorrectly in my edge animate comp. after fixing this, everything worked!