Unfortunately, some HDMI cameras output their audio in a way that SlingStudio is not compatible with.
You can connect these cameras using an HDMI cable (which should carry both the highest-quality video as well as the highest-quality multichannel audio) but it doesn't make any difference: The video portion may look fine, but not audio comes across.
You should try the usual troubleshooting steps first (for example, make sure the camera is properly connected, try a different HDMI cable, make sure that the camera doesn't have some audio-out setting that could be causing a problem). But if none of these steps makes a difference, then you may have an HDMI camera that doesn't properly output audio signals to SlingStudio.
In that case, you'll need to explore other options for audio. For example, you could use a nearby smartphone for audio instead. Or you could use a preamplified microphone that's connected to the SlingStudio audio-in port (directly above the AC power port).