Here is the FAQ for Chromebook:
You probably want to pay attention to the question "I use a specific piece of traditional software. Will it run on a Chromebook?"
Keep in mind the Arduino IDE is made up of several tools which all would need to be supported. Plus, device drivers for the underlying OS would need to be supported as well. (e.g. the Uno emulates a USB Modem, it is unlikely the Chromebook's OS would come with a device driver for a USB Modem.)