In most cases, you will not need 2 microphones. Both Windows and Mac operating systems should allow you to use the same microphone in the Chrome browser and in your RSI event. You should also be able to use the same microphone in 2 windows/tabs of the same Chrome browser or even in different browsers. You will be able to mute and unmute your microphones completely independently.
However in some rare cases because of hardware limitations you have to use 2 separate microphones.
Please test your audio set up in advance.