If you want to use the Edison on another PC, you won't be able to use the Arduino IDE without installing the Edison's driver. If the computer has the FTDI driver installed by default, you will be able to access the Edison's console, where you can install mraa (if you have access to the internet on your board) and use it to compile code in C, Python, Node.JS, etc. If the FTDI driver is not installed, then your last option would be for you to SSH into your Edison and you will be to also access your Edison's console. Nevertheless, accessing your board's console through a USB port or through SSH will require to have PuTTY or any other similar software installed.
So, I believe that your best option would be to find a portable version of PuTTY (or any other similar software) so you don't have to install anything and access your board through SSH.