It is totally possible (watches with gravity-powered generators are widely available). I think the question would be whether it produced enough energy to justify the cost and added weight.

If you've ever used a flashlight powered by "shaking" you'd know what I mean (they have an extremely low light output vs. the amount of shaking needed).

I'm not sure about the wattage considerations but I assume it would be complicated to transfer such a wide range of power to the battery.