I'm not sure that a 16bit add-on card is needed, rather an Ethernet chip. Then you'd have to select one of the well know streaming protocols and implement it on the Teensy.
At the very least, you'd need the Wiz820io Ethernet modules. The simplest way to connect them to Teensy 3.2 is with this adaptor board. Or you can just connect 5 signals and power+ground.
Software-wise, this may not be easy. You'll have quite a bit of code to write. The audio and ethernet libraries can do most of the tough low-level work, but you'll still have to fill in quite a lot.