My suspicion is that the artifact you describe is caused by the diffraction of light around the mask that defines the edge of the frame inside the camera. This is normal and impossible to avoid completely. Most people never notice it because it is cropped out during printing, or hidden by slide mounts.
If you look closely, you will see that it occurs to some extent along all the borders. (Though it may be only a few pixels wide.) It will be more or less apparent depending on the scenes you are photographing. For instance, I would expect it to not be apparent in completely white or black frames.
If you look at the second sprocket scan in a window with a dark background, you can see a similar effect along the right border (but in this case, it is a scanning artifact). Perhaps someone else knows more definitively?
Loose Pressure Plate?
The effects of diffraction may be worsened if the film is not held completely flat. You can look at the pressure plate of the camera to see if this is the case.
The first image you provide also shows what appear to be reflections outside the frame along the bright portions of the top and bottom borders. I think these were produced during scanning, though they could be from the camera as well.
The current scans with sprocket holes are at too low resolution to evaluate for these reflections. Higher resolution scans that include the sprocket holes would be helpful.
Another answer states that a malfunctioning shutter might cause such a problem. I doubt this is the case, but if you're interested, you can see the shutter of a DSLR in action YouTube: Inside a Camera at 10,000fps
Shutter problems can be ruled out by exposing images independently of the shutter. This can be done by covering the lens with several layers of thick black cloth. Then expose the film by removing the cloth while the camera is in bulb mode. Replace the cloth before closing the camera shutter. A tripod and neutral density filter would be helpful.
Take an entire roll of photos this way. If even just one image shows the white line, it is definitely not caused by the camera's shutter. However, not seeing the white line in any images does not prove there is a shutter problem.
Since this is an unusual way to expose film nowadays, you might get some interesting photos in the process.