If you're sure the wiring inside the cord isn't damaged, you can probably make a simple repair to the insulation. Zip cord is typically made of a vinyl (poly-vinyl chloride) material and you may be able to find a solvent, like for PVC "plastic" plumbing, that would dissolve the insulation enough to reform it to seal the tear, if there isn't a chunk missing. If there is material missing, perhaps a vinyl repair product would also work.
Since you made a point of mentioning it was vintage, I think these suggestions will give you the most attractive resolution.
I've seen chewed cords and this one looks like it had been caught under a piece of furniture or closed in a drawer. The stretching doesn't suggest teeth marks.
Leaving the cord alone shouldn't affect the operation of your device. Rather, it's a safety issue. Electrical tape would prevent contact with the wiring, but it's not an attractive option. Using a vinyl adhesive or patching product would restore the integrity of the insulation without compromising aesthetics.
If the wiring is actually damaged, though, replacement or shortening the cord to removed the damaged area is best.