So I am trying to make a digital keyboard on webpage, that when one of the keys are clicked it creates sounds on the Arduino board. I want to do this without an ethernet shield. I am a beginner so I am not sure how to proceed. I'm not even sure what type of code it needs to work. I just know that I need Processing sketch to act as a go between for the webpage and the Arduino.
You are asking for something that sounds simple, but it's far from simple.
The security of modern web browsers is based on simple premise. Websites cannot in any way interact with the system of the user. Unless you are a hacker ant you are using an exploit you cannot make a website that will send data to the PC it's opened upon, or even read anything (like files and folders from hard drive) except the things that browser tells it (like what fonts are installed)
You can write a browser plugin that will break that rule, but it's a bad practice to deliberately make suck an awful backdoor.
The arduino code will be the simplest task in your project.
The code should read from serial a string value. Since You can only press one key at the time with a mouse you just need to read a value of key that was pressed like "12". Then play correct sound using for example a buzzer
The ethernet shield can help only by allowing you to install server on the hardware itself, connecting it to your wifi and forwarding it outside to be accessible by everyone on the web. For that I suggest using raspberryPi, omega2 or even esp8266 and not an arduino.