Yes, any microcontroller would work.
The LEDs, you can drive directly from the microcontroller, it usually can output enough current on each IO pin to do that. If you want to use a microcontroller with fewer IO pins, then you can use a led driver that can control the number of leds you're planning to use (there are shift register like led drivers that have 16 channels, so you only need two wires to transfer 16 bits to the driver, one bit for each channel you want to turn on or off
You could also use addressable rgb leds where you can basically connect the rgb leds in a series of 6 leds and then each time you need to update a led, you send the data for all leds again through a single wire.
You don't power relays directly from the microcontroller.
You power the relays from 5v or 12v or even 24v, whatever is available in your device. 12v would be preferred, as it means less current will be used in the primary coil to keep the relay energized.
You use npn transistors or n-channel mosfets to close the circuit and make the relay work, and you control that transistor or mosfet with 3.3v - 5v from your microcontroller pin.
There's chips like ULN2003A - 7 channel darlington transistor array - basically 7 on/off switches you can control directly from your microcontroller. These are cheap and will work with 12v relays well, but won't work well with 5v relays because there's a voltage drop of around 1v inside the chip, so the relay's coil would receive only around 4v, which could be too low for the relay to work reliably.
For 5v relays, you could use the mosfet version of the chip, ULN2003
V12 -
https://www.digikey.com/short/ztp997jjAs it uses mosfets instead of darlington transistors, the voltage drop on each channel is very low so it would work with 5v relays.
All the above are lo-side switching (the relay's primary coil is connected to ground to close the circuit and turn on the relay.
ULN2803A mentioned above would also work but would be used to source, as hi-side switching, it gives voltage to the coil of the relay and the other side of the primary coil is connected to ground by default. It's less common method to control relays.
for flipping stereo channels , look into analogue switches ... you could use them to switch one input between 2 or more outputs with minimal added resistance.
So for example by default each channel would be connected to output 1, and if you want to flip them , you switch both inputs to output 2
channel 1 (L) ---- L ---- R
IN1 OUT1.1 OUT1.2
channel 2 (R) ---- R ---- L
IN2 OUT2.1 OUT2.2