How can I create a rotating bicycle chain in AE from a vector imported from AI

Here's is what I would do. Open Illustrator. Put the top and bottom straight sections of the chain on separate layers and make each length at least one link longer. Make the front and back circles complete with links all the way around and put each on a separate layer. 

in After Effects create a track matte for the top and bottom straight sections that end at the tangent point of each circle, animate the top chain position in one direction and the bottom in another so they move exactly one link length. Now animate the rotation of each circle far enough to move each exactly one link length so the edges match up. Add a half-circle track matte for both circles of the chain so the inside links are not visible. 

Now trim all layers to the matching keyframes. You now have a one-link move animation. 

Select all layers, pre-compose trimming the new comp to the length of the layers, add Time remapping to the Pre-comp. Move to the last keyframe for time remapping, then move back one frame, add a new frame, then delete the last keyframe. Add a loopOut() expression to time remapping. 

You now have a continuous rotation chain that will last as long as you need it to last. 

Pre-compose again, enable time remapping, and add some keyframes to start and stop the movement of the chain. 

There you go. I don't have time to post examples, but that's the efficient workflow for this kind of thing.

No need to overthink it. Simply duplicate the animated layer and stagger them suitably.

Mylenium

You would simply import a single link, then copy & paste the path to the position of the link, fiddle around with the timing and the "orient to path" option and then clone the link with temporal offset. You'd do so in a pre-comp so you can use time-remapping to control the speed later on and repeat the motion as often as needed.

Mylenium