Bicycles have serial numbers, but usually don't have VIN's at all.
There is a chance of some confusion here--because with a vintage or antique example of a motorized bicycle, it may be possible to register it without a VIN, or it may be possible to request a VIN for it (IL is one state that will grant VINs to antique/vintage vehicles). States write their laws so that if a particular vehicle was ever allowed to be registered in the past, then they will always have provisions for it to be re-registered--but they won't necessarily allow you to construct a new example and register that.
When you e-mail your dept of motor vehicles about motorized bicycle laws, you need to be very specific: you should ask them about taking a new bicycle and adding an engine yourself. You should not ask simply about "motorized bicycles", because there are probably provisions for allowing registration of vintage and antique examples of motorized bicycles (previously registered!) that your new-built motorized bicycle will not qualify under.