your right you cant short the two signals together.
what you really need is a line in on one of the cards and connect the line out from the other then the built in mixer can handle it. I do this with a audigy 2 and a vortex.
i think for external hw using two analog to digital converters and then you can combine the two numeric values and use that to output out on an analog output.
but i think that would still likely need some type of a microprocessor. maybe an rpi with a i2c sound modules.
but make sure which ever way you go the outputs are not amplified.

you can't "just" short the signals together, but you can if you put resistors in between, that's essentially what a passive mixer is, just with variable resistors, and you can also get those ready made for cheap in a very small form factor.

obviously you should make sure that all the devices are roughly at the same signal level though, and I'd certainly use line level for that and not an amplified output, mostly every sound card should have a line out.

The most easiest functional way is this :

Use a 10kohm pot per signal pair and keep the output wire short, impedance of the signal is much increased and it will easily pick up noise and interference compared to a proper active design.