How do I connect a PS4 controller connected to an Arduino that's connected to a laptop to another Arduino? My high school is entering an ROV competition and for next year's competition we want to try and set it up so that we can control it using a PS4 controller. I have done research and found this shield for Arduino and this library that allows you to connect it to an Arduino. I plan to then connect the Arduino to a computer via USB and then send each button press to a Processing program via serial. Would it then be possible to send this from Processing to another Arduino out of the laptop? I'm just wondering how feasible this project is.

We initially had just a single Arduino at the top with the Ethernet shield and took inputs from standard joystick potentiometers. We want to add in using the PS4 controller this year for easier control. We found the USB Host Shield that gives us that capability, but uses the same pins as the Ethernet shield. So we added another Arduino, connecting the laptop as the bridge also allows to have a GUI. The Ethernet is wired, by the way. We need the laptop, it's not just for a bridge.
this is possible, but the processig bridge can add a lot of delay if not correctly handled, also wifi has a low range, almost 100m by default hardware.

zigbee module has better range (xbee pro are up to 1km) and are microcontroller friendly, you use the as you are using a serial, they will hide their complexity once you set them up, and this can be done by PC.

That mens less code, that mena less error and more ram/flash/cpu so you can do a smarter robot :)

zigbee use the serial protocol to communicate, exactly as you do with the pc. Also arduino can use, with the library SoftSerial, any pin as Serial. So you can keep the HardwareSerial to communicate with pc, and any 2 free pin to communicate with the xbee. So you can use the arduino stand alone, only with the controller, and when needed connect the pc. Then arduino can act as a bridge between xbee and pc, so pc can excange data with the rover. But even without pc your project will be fun.
The same apply for bluetoot or 400/800mhz rxtx module, but not for wifi shield