Make Piano keys on Webpage create sound on Arduino (w/out Shield)
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.

Websites can only communicate with the server they were opened from. You can use AJAX made easy by a javascript library like jQuery. This means that You will have to write your own plugin to any of the existing http servers like apache (for php), or create your own server using for example flask framework (for python) and use it to communicate through serial protocol with the hardware.

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.