When Gina Wagner’s 8-year-old daughter Bronwynn received an American Girl doll as a birthday gift, her son, Miles, 6, began to ask whether he’d be getting an “American Boy” doll for his next birthday.

It may not seem like it, but doll making and repairs is a giant segment of the craft and toy industry. To reinforce this, simply visit an antique shop that specializes in antique dolls or a retailer ...