Aperture can have an effect on barrel or pincushion distortion in lenses. Only points on the optical axis see the full aperture (the pupil). Points away from the axis see only parts of the pupil, mostly in an asymmetric way. This is what leads to vignetting. The beam coming entirely from one side of the pupil at the image corners also moves inwards or outwards with aperture. This is what causes the distortion to change.
The effect is most pronounced in wide aperture lenses that have strong vignetting and focus shift.
If you look at the beam of a wide aperture lens at full aperture vs. the beam at small aperture (can't seem to find a proper link right now), you'll see what I mean more clearly.
Disclaimer: I work with astronomical instruments for a living, same principles apply even though we don't put apertures on out telescopes ...
EDIT: After coffee, I realize this works perfectly well without focus shift at all, it is simply the fact that different parts of the pupil form their image at diffrent distances from the image center (distortion). At large apertures, vignetting takes care of eliminating most of the pupil, leaving a half-decent image, at small apertures it is the aperture ring that cuts out most of the pupil leaving only the center, forming again a single decent image with different distortion.
- N.