Is each channel a dedicated stream? Do you care about channel 1 stepping on channel 2? If yes, you will want a dedicated radio for each channel. If you already have radios for each channel, you could use Rx out from a radio. Other solutions would be two scanners and a PC. The most cost effective solution is setting up an SDR solution if you do not have scanners or radios dedicated to the channels already. Trouble with the SDR solution is that it will take more technical aptitude to setup.
As for the PC side of things, if not using an SDR, get yourself some cheap USB sound cards (or similar device) for each stream. The goal being to have multiple line/mic inputs so each stream can have a dedicated audio channel unless you are running an SDR. If you use an SDR you do not need external sound cards. The computer doesn't need to be anything fancy if all it's doing is streaming audio, enough power to run the apps you intend to use to stream with and other management tools.
With all of that in mind, I suggest you choose an audio source first. Pick how to get that audio to the computer, whether it be SDR and the audio stream is handled by the computer or the audio stream is an external source and you need multiple sound cards. If you use a radio, you only need to worry about audio. If you use scanners, you could look into setting up a program to control and program the scanners in addition to Rx audio. Once you have your audio sources you need to pick a solution to stream the audio.