Although it is customary to refrain from haircutting during Sefirat Ha’omer until after Lag Ba’omer, Ha’Rav Ovadyah Yosef (a”h) stated that this custom applies only to men; women are allowed to take haircuts during the weeks of Sefirat Ha’omer. Regarding children, Ha’Rav Ben Tzion Abba Shaul (a”h), in his sefer, Ohr Le’Tzion writes that since the prohibition against haircutting applies by force of Minhag and not the strict halachah.
Therefore, young children may be given haircuts during the Sefirah period. Children who have already reached the age of chinuch (ages 5 or 6) should preferably not take haircuts during the weeks of Sefirah, but even regarding such children there is room to be lenient when a need arises. Therefore, if a child who has reached the age of chinuch and has overgrown hair and looks unkempt, may take a haircut during the period of Sefirat Ha’omer.