The problem with VCAs and voltage-controlled resistances is achieving perfect tracking of the gain in each channel. You don't say how accurate you need the channels to be matched to each other; if somewhere around 0.5 dB variation between channels is acceptable, you have a wide range of options to choose from.
The SSM2018 mentioned in another answer is very very good, but its datasheet notes (p.11):
A 25°C temperature change causes a 8.25% increase in the gain
constant, resulting in a gain constant of 30 mV/dB.
The much cheaper (dual) LM13700 features 0.3dB gain tracking between different amplifiers according to its datasheet.
However if you need much better gain matching between channels, take a look at an MDAC - Multiplying DAC - which is a switched attenuator, with the gain setting applied by changing the code on the digital inputs. So you would need to read the pot in the ADC built into a microcontroller and write the appropriate code out to the MDAC for all channels.
The gain matching achievable is limited by the precision of the MDAC, and you may be able to achieve higher bandwidth than devices specialised for audio use.