J.R.R. Tolkien’s close friend and fellow Inkling who wrote The Chronicles of Narnia was C.S. Lewis.
Tolkien and C.S. Lewis were colleagues and friends at Oxford. They belonged to the Inklings, an informal literary discussion group that met mainly at The Eagle and Child pub and in Lewis’s rooms at Magdalen College. Members read drafts aloud and discussed literature, mythology, religion, and language.
Lewis encouraged Tolkien’s fiction, and Tolkien in turn influenced Lewis’s intellectual and literary life. Their friendship was important, though it also experienced periods of strain. Both writers became famous for modern works rooted in mythic and Christian themes, but their projects were different: Tolkien created Middle-earth, while Lewis created Narnia. Charles Williams and Owen Barfield were also Inklings, which is why they can seem plausible in this question, but neither wrote The Chronicles of Narnia.