Hi,
I am trying to set up ProgDVB with the Media Server with a dual satellite card (TBS6981)
Both inputs are fed from the same satellite.

I want to be able to select 1 transponder from each input and then, using Media Server, broadcast 2 or more channels from each transponder.

So, for example:
Transponder 1 on frequency 10818 broadcasts its channels 1,2,4
Transponder 2 on frequency 10714 broadcasts its channels 1,2

Is this possible? I can only get 1 transponder to work at a time.

Yes. I am recommend setup 2 copes of ProgDVB ot different folders and setup 2 media server for 2 transponders.