The PC doesn't really know or care how many keyboards are connected or where each keypress comes from. You can type "ABC" by pressing "A" on one keyboard, capslock on a second keyboard, "b" on another and sending "C" from an Arduino.
If your Teamspeak can only accept keys on a keyboard as input and you can't type keys at the same time as doing whatever you are doing then you are out of luck. There is no solution to that. You need to search for another way to make it work.