* Andy's Mom: She's shown to be caring and involved in the lives of her children. It's plausible she would handle repairs for toys, especially Woody, who's Andy's favorite.
* Andy: While he's a child, Andy shows genuine affection for Woody and likely wouldn't want him to stay broken. He could try to fix him himself or ask his mom for help.
* A Local Toy Repair Shop: It's possible Andy or his mom would take Woody to a shop that specializes in fixing toys. This would be a more professional solution but might not feel as personal.
Important Note: The Toy Story movies don't explicitly tell us who fixes Woody. This leaves room for fans to imagine and speculate!