The nature of the repair and who would be best to do the repair depend on the nature of the damage, the material the speaker is made of and the type of finish. Without a picture it is tough to say but you may want to look for a guitar repair shop if it is just finish damage. I would expect guitar repair shops to be more common than piano repair. Fixing a finish defect is probably more like repair of a guitar finish than fixing a cabinet or furniture, unless the speaker is MDF and there is damage to the underlying substrate. A picture would be really helpful if you can figure it out.
To get some ideas you can go watch a few videos at Stewart McDonald such as
Fixing a small chip in your guitar's finish - StewMac