API integration with AzureCast is a pretty broad scope of work. I’m assuming the kind of integration you want with AzureCast in Discourse is a onebox that links to a radio station?
Which doesn’t appear to be working. A onebox embed should be able to have a little audio player like this in a Discourse topic:
Looks like AzureCast has support for embedding:
Public Pages: AzuraCast includes embeddable public pages that you can integrate into your existing web page or use as the basis for your own customized player.
So something like this should be possible like we have for soundcloud: