Prior research has demonstrated the importance of children's peers for their learning and development. In particular, peer interaction, especially with more advanced peers, can enhance preschool children's language growth. In this paper, we explore one factor that may modulate children's language learning with a peer-like social robot: rapport. We explore connections between preschool children's learning, rapport, and emulation of the robot's language during a storytelling intervention. We performed a long-term field study in a preschool with 17 children aged 4–6 years. Children played a storytelling game with a social robot for 8 sessions over two months. For some children, the robot matched the level of its stories to the children's language ability, acting as a slightly more advanced peer (Matched condition); for the others, the robot did not match the story level (Unmatched condition). We examined children's use of target vocabulary words and key phrases used by the robot, children's emulation of the robot's stories during their own storytelling, and children's language style matching (LSM—a measure of overlap in function word use and speaking style associated with rapport and relationship) to see whether they mirrored the robot more over time. We found that not only did children emulate the robot more over time, but also, children who emulated more of the robot's phrases during storytelling scored higher on the vocabulary posttest. Children with higher LSM scores were more likely to emulate the robot's content words in their stories. Furthermore, the robot's personalization in the Matched condition led to increases in both children's emulation and their LSM scores. Together, these results suggest first, that interacting with a more advanced peer is beneficial for children, and second, that children's emulation of the robot's language may be related to their rapport and their learning. This is the first study to empirically support that rapport may be a modulating factor in children's peer learning, and furthermore, that a social robot can serve as an effective intervention for language development by leveraging this insight.