No experience with the Sony, but firsthand experience with the Canon VIXIA RF H80 for livestream use. Requires separate power via the included AC power adapter. Otherwise it works well.
You can also use a DSLR that's video capable. Lots of used stuff available out there for decently cheap prices.
For sound, is there a monitor output on the sound system? A headphone jack will work in a pinch, though you have to be careful not to overboost the signal. That's the best way to capture sound. Monitor Out > Camera External Mic Input. Onboard mics just don't work very well unless you are standing in direct proximity of the camera.
Keep in mind you'll also need a video encoder, either standalone or as a peripheral/card. Some video cards have built-in encoding. Check your computer's specs to see.
Brad