Can you turn a non-USB audio mixer into a USB mixer?
Essentially, USB audio devices are all simple USB sound cards/adapters. Most common ones have analog stereo in, but plenty have multiple channels input as well. Most use the default USB audio drivers. Some have higher quality ADC with better codecs for a better audio. 

Since you only need an audio input into the computer, and no automation or computer control of the mixer, just find a decent USB sound card with multiple inputs (or multiple USB devices) and a good audio software.