How to animate roller chain with three or more sprockets?
I would constrain the three sprockets so they only have one degree of freedom (an easy way is to mate the hole with a pin/screw and then constrain it to a block or something) then it should move the chain when you rotate one of the items.
Hope this helps,
See if this helps constrain the chain to the sprocket:
Also, you could use a belt and wheel to make the animation much easier or you could adapt the same procedure used for 2 sprockets and constrain the 3rd the same way.
Hope this helps