There is no real unified parts list for custom built portables like this, but I can list some stuff you'll need.
- Hot glue or quick set epoxy. Things will need to be held in place. I recommend using thin 3D printed brackets if you can to keep things neat.
- 22AWG stranded wire for power delivery. Preferably 7 different colours, including red and black.
- 30AWG PVC jacketed kynar wire. As many colours as you please, 4 is probably a good number. For data lines and USB.
- 32-38AWG enamelled magnet wire. For small and fiddly via soldering as well as BT module if you want to relocate it.
- Voltage regulators. Currently the RVL-PMS-LITE from https://4layertech.com/ is the easiest and recommended method. It also has integration with the USB-PD2 module that enables plug and play charging of the system via USB-C cable as well as data transfer to and from a PC using the USB-C cable as well.
- Audio amplifier. Currently the best one is the U-AMP also sold by https://4layertech.com/ that has an optional integration with RVLoader to allow changing the output volume via controller button combo. There are also several cheap no frills amps available on ebay.
- Controller board. Again to https://4layertech.com/ the GC+2 is an advancement of the original GC+ once sold by the now defunct BitBuilt store. It functions exactly like a regular Gamecube controller, and has handy solder pads for all buttons and joystick lines as well as a secondary Z trigger for compatibility with the GC2Wiimote patch.
- Cooling. The most commonly used comact cooling method for portables is a 35x35x7mm heatsink + an identically sixed 5v fan with a solid metal base attached to 70x35x2mm copper plate via thermal paste or thermal epoxy. The plate distributes the heat across the entire volume of the cooling assembly and is one of the more effective methods available. Some people have also used cheap flat all metal 5v CPU coolers before. They work, but they're quite large in comparison.
- Buttons. The Gamepad you're using has pretty much all the buttons you'll need already, but you will need some dual layer tact swtches for the LR triggers and probably a few hard tacts for the face buttons below the screen. I can't remember the part number of the recommended dual layer tacts, so you'll have to ask someone else for it.
- FFC breakout. The gamepad's inbuilt controls use carbon traces that can't be soldered to, so you'll need an FFC breakout to plug the gamepad control FFCs into to then be able to solder wires to the GC+. IIRC the pitch is 1mm, if you count the number of pins on each FFC, you can probably find some premade FFC breakout boards on ebay.
- Rumble motor. Rumble is optional, but if you want to have it I recommend using replacement Wiimote vibration motors. They're small, stronk, and run on 3.3v.
- No clean flux. Get it, you need it.
- Kapton tape. Super flat electrically insulating tape useful for preventing PCBs from shorting on each other. Also good for securing wires.
That's just off the top of my head, I may have missed a thing or two. I live in Australia, so I can't really help with locating specific products abroad aside from "check Ebay, Amazon, Digikey, Mouser"