This article might help:
https://azure.microsoft.com/en-us/documentation/articles/key-vault-use-from-web-application/
Note, since you are using an API App, you have to get to the underlying "Host" to see the functionality available to Web Apps. Go to the API Apps blade in the portal, and click the "Host" link at the top of the blade. Then,
you will be able to follow the steps in the above article.
Thanks,
Mohit