I didn't see anything about this on the Replay web site, but does this software require the "System Mixer" to be available and use that as the source for recording? Usually, the System Mixer (could be called Stereo Mix in some systems) is required to be able to record streaming audio.
Check the Recording Devices in the Control Panel, Sound Panel. The System Mixer should be set as the default playback device. With the system mixer, whatever you hear on the PC speakers can be recorded.