How do I get a new chainring when my bike is so old, the manufacturer doesn't have the chainring (or crank) I need I need a new chainring but when I called the manufacturer they said my model is so old they don't have the part any more.

I've attached an image of the crank from the manual and a photo from my bike. Can someone tell me if it is possible to get this type of chainring, or an entire new crank? 

How do I measure it to make sure it fits my bike?
It seems that you're asking about a BMX chainring. As far as I can see (I'm no expert in BMX) they are fairly standard. To be sure you'd neet to take three measurements:


the diameter of the hole that goes on the bottom bracket spindle (1)
the diameter of the hole that attaches to the drive-side crank (2)
the distance from the centre of the first hole to the centre of the second hole (3)


Have a look at the attached image with those measurements marked red:



As you can see, the measurement 3 is somewhat flexible with the chainring pictured.

With those three measurements you can visit couple of LBSes and confirm that you're purchasing the correct part.