Kundalini awakening as a child is a rarity yet possible. Lord Krishna, an avatar of his era had his Kundalini awakened fully when very young probably 2 to 4 years of age. The reason how Kundalini awakened fully when only a child laid in philosophy of spirituality! The reason attributed to this is the kundalini was already quite awakened in the previous manifestation. Before the kundalini could realize its full potential, death of the human form occurred.
In the next manifestation as ordained in spirituality, the human form manifested Kundalini power from the very same level left in earlier life. On path of pure spirituality nothing ever goes waste. Every single development in the field of spirituality keeps accumulating. The human being keeps evolving to higher states of consciousness, prime reason why Kundalini of Lord Krishna was almost awakened when born.
In the case of Adi Shankaracharya who completed his commentaries on Bhagavad Gita, Brahma sutras and Upanishads quite young in age, the Kundalini again was awakened early.
Any seeker, traveler of spiritual path working for welfare of mankind all the time can awaken once kundalini early by practicing absolute celibacy for a minimum period of 12 years in continuation.
For kundalini to reach its full potential, God Almighty ordained 1.1 million manifestations in human form, a long journey indeed! To awaken it at a young age is nothing short of a miracle. Most preceptors like Mahavira, Jesus Christ; Adi Shankaracharya even Sri Ramakrishna Paramhansa and Maharshi Ramana awakened their kundalini early.
Swami Vivekananda also awakened his Kundalini almost fully but was not prepared to reach God in his lifetime. His goal of life was different, community service. Full awakening of kundalini by Swami Vivekananda was reserved for next manifestation.