Although it is customary to refrain from haircutting during the Sefirat Ha’omer period (until after Lag Ba’omer), Hacham Ovadia Yosef rules (in Yalkut Yosef, p. 76, and Hazon Ovadia, p. 261) that this custom applies only to men; women are allowed to take haircuts during the weeks of Sefirat Ha’omer.
Regarding the status of children with respect to this Halacha, Hacham Ben Sion Abba Shaul (Israel, 1923-1998), in his work Or Le’sion (vol. 3, p. 184), writes that since the prohibition against haircutting applies by force of Minhag (custom), and not the strict Halacha, young children may be given haircuts during the Sefira period. Children who have already reached the age of Hinuch (education) – generally assumed to be around the age of five or six – should preferably not take haircuts during the weeks of Sefira, but even regarding such children there is room to be lenient when a need arises. Thus, if a child who has reached the age of Hinuch has overgrown hair and looks unkempt, he may take a haircut during the period of Sefirat Ha’omer. This is also the ruling of Hacham Ovadia Yosef, in Yalkut Yosef (p. 184).
Summary: Women and young children may take haircuts during the period of Sefirat Ha’omer. Children who have reached the age of Misva training (five or six) should not take haircuts, but they may in situations of need.