Small engines typically had a magnet for the magneto
(newer ones, I dunno) and you could install a second
magneto to pick up power. However it will be fairly
limited current and high backing voltage, unless you can
shunt (rectify) it into a nice big reservoir capacitor.
There may be a substantial reverse spike too, which
might need snubbed or bridged. The magnetos have
the unfortunate property of having one end case-grounded
so bridge rectification is likely a no-go.
I don't think there is a whole lot of power there, probably
not enough to run a high impedance injector.
Needle valve sticking is a hygeine problem, not a
mechanical one. Why you want to fix "gum" with high
tech gizmo action, I don't get. Nor why you think a
highy speed cyclic moving needle valve is going to
fare better than a quasi-static one. From messing
with EFI for a while, I kind of doubt it - I've had
carbs last longer than injectors especially in bad
fuel quality situations.