MADURAI, India—The future of India’s booming e-commerce market is in the hands of small-time customers like 27-year-old Gayathri Rajamansingh.

Each Sunday, the owner of a small hair salon browses the Shopclues website from her home, hunting for bargains. Recently, she fixed on a floral-print sari, a traditional Indian one-piece garment, and clicked “Buy Now.”