Some ideas...
Hi Stew...
I can't really tell from your photo, but it appears that the veneer is thicker than the typical thin, wood-grain vinyl that you can buy as peel-n-stick rolls. Yours appears to be breaking into fairly solid pieces.
If it's thin and flexible, you can get the rolls of wood-grain lookalike stuff at hardware stores. It's usually called contact paper, and is used for covering shelves, etc. There is usually a choice of different colors, etc. You would need to remove the old from the edge, fill in any missing backing material to provide a nice flat surface, and recover with the new material.
If it's thicker and stiffer, like yours appears to be, laminate may be the best option. The brand most likely to jog your memory is Formica. Good building supply stores should carry a good selection. It would probably be better in the long run to remove the entire broken edge or section, and replace with new. I'm guessing it will be much easier than trying to cut and fit a bunch of little repair pieces. The challenge will be that laminate is usually thicker than your veneer.
A final option is one you may not have thought of: real wood veneer. I think this would absolutely be the easiest solution, and give you the best looking final results. I know that Home Depot carries rolls of real wood veneer, usually white oak. They are about an inch wide, thin and easy to use, and unfinished so you can easily stain them to match your existing colors. Best part, they're easy to cut to fit with an exacto knife or other sharp knife, and you could glue two strips side-by-side if they weren't wide enough to cover the damaged front edge of your speaker. Also, do a google search on wood veneers and you'll find several companies that sell walnut veneers in every size imaginable.
Good luck with your little fix-it project, and let us know how it worked out!
Clay