KAKINADA, India—Entrepreneur Ajay Sinha made stuffed toys in China for a decade before he started doing something almost unheard-of in his industry: manufacture in his native India.

At Mr. Sinha’s new factory here recently, dozens of sari-clad women assembled Elmo dolls for Hasbro Inc., cutting furry red fabric, running it through sewing machines and stamping plastic eyes into smiling faces using a hydraulic press.

As...