I'd think you need a tiny DC motor that includes a gear-reduction on the output. You probably want the mechanism spinning at, say, 100 RPM or so. Without gear-reduction a typical micro motor will spin at thousands of RPMs. Micro DC motors with a gear-reduction output are only a few dollars on eBay but I'd think most might be a tad longer than the AC-vibrator-motor housing on the windmill.
So as depicted in the bottom photo above, mechanical modifications would be required.
The task of converting the existing AC voltage to a suitable low-voltage DC to drive the gear-motor is the "easy" part - another few bucks for an eBay voltage regulator module which can be identified should you go this route.