These things aren't designed to be repairable. Electronics repair these days amounts to basically removing whichever circuit board is bad, and replacing the entire thing.

And given that places like GameStop are practically giving away any GBCs they may still have in stock, heck, they're practically giving away any GameBoy Advances they have, it just doesn't seem worth the bother. Even if you managed to find someone selling theirs for parts, there's likely really only one circuit board in the GBC, so if theirs is broken, it won't do you any good.

I'd say go to a local GameStop or online, just get a GBA. They play all GB/GBC games, and you can expand into GBA games.