I've done a nearly identical project as a one-off for a customer in the past.
Motorised Potentiometers exist (ALPS is one manufacturer), but the response speed is slow, so that may not be suitable for your application.
Digital Potentiometers are available from multiple manufacturers and are generally SPI/I2C controllable, which are both easy to interface to your controller of choice.
Digipots do have a couple of serious limitations that need to be considered for this sort of project.
They have a maximum voltage window between different terminals (usually, but not always, the device supply voltage) and requirements for the ground reference. This may or may not be a problem depending on how your existing device uses the potentiometers.
For my device the maximum voltage differential was within the acceptable range, but the ground reference was not suitable. So, I used a small isolated DC-DC converter (e.g. XPPower), and a isolating-coupler (e.g. TI iCoupler) to allow a Digital Potentiometer (e.g. Microchip) to be placed on each channel while staying within the device limitations. A low-cost microcontroller drove each channel via SPI.