Um, this is for everyone who uses firefox and cannot get coconut shy to work, u can go to this page and play it ^^ http://swf.neopets.com/halloween/coconutshy_v6.swf
I've been using this method since forever and it's the only reliable way to play that I know of. You gotta remember though, it is a carny type game. In other words, it's rigged just like a carnival game. It isn't going to let you knock the coconut down, or even hard enough to shake it real good, each and every time. It will fall eventually though! Oh yeah, if you will put your cursor on the 1st "a" in the word 'again', where it says "PLAY AGAIN", this is where it needs to be.
You can also just right click to get the flash player drop down thing up, then click on a coconut. A lot of people think that the darkest ones are more likely to fall. Edit: Also, no method makes it fall. It's random whether it barely moves, shakes, falls, or explodes. I don't think the OP meant that it was for making it fall to get the avatar, just that it makes it faster to play - meaning you won't be spending as much time lining up, and throwing. Just do it every day, and you'll eventually get the avatar.
This method works, but even the slightest movement will make it miss. You have to aim directly for the center