You can send audio signals from Ableton Live to MadMapper for audio reactive things - LED fixtures can then be audio reactively synced.
Option 1: you can do an hardware audio connection from your audio mixing hardware to your computer line in.
Option 2: you can connect them with software. If your audio output from Live goes to an audio card, you can change it to a virtual audio output that would be router to both your audio card and MadMapper.
On macOS you can use apple "Audio MIDI Setup.app" for that (in Audio Devices, press the + button at the bottom and choose "Create Multi-Output device" (google for more info), on Windows you have to install a virtual cabling software (but you then loose the possibility to use ASIO driver to your audio card directly, hardware cabling might be better...).
there's a program called "Virtual Audio Cable".
It allows you to set the output of your music software to the virtual cable and then set your main sound input to the mic.
On macOS, SoundFlower & some others exist.