Is it possible to save my progress online in Candy Box 2?

Unfortunately you cannot save to the server but you can transfer your saves to different machines.

You need to save the text by clicking this button

You can then load it on any machine by pasting the text in the box below.

No. Save data is stored locally in your browser cache, and can only be transferred to another computer using the "export save" function. It is not stored online anywhere.