Audio interface or External sound card?
Sound card == audio interface

The only difference is that the name "audio interface" is usually used for pro audio equipment. On the other hand, "sound card" usually refers to consumer audio.

Currently there are no cheap multi-input audio interfaces. You may want to check out second hand market, but even there USB and 1394 interfaces are expensive.

The only cheap solution that comes to my mind is to use multiple stereo-input interfaces and synchronize theirs clocks using SPDIF. Then you can link them on Linux (in asoundrc) or Mac OS. I don't know if it's possible on Windows. 

However, I have never tried synchronizing clocks using SPDIF myself.

Without synchronizing clocks you need to use resampling which is computationally expensive and can degrade quality. alsa_in and alsa_out tools for Linux/JACK can do this. But they're usable only if you don't need accurate timing as they don't do latency compensation.