According to the Palo Alto Medical Foundation, underarm hair starts growing about two years after pubic hair develops. The age that this happens varies somewhat between females and males.
The Palo Alto Medical Foundation indicates that pubic hair typically shows up in girls between the ages of 8 and 14, so underarm hair appears between the ages of 10 and 16. For boys, the growth of pubic hair generally begins when they are between 10 and 15 years old, so underarm hair emerges sometime between the ages of 12 and 17.
KidsHealth discusses "precocious puberty," where signs of maturity are seen in girls before age 7 and in boys younger than 9; with these children, underarm hair often appears at that time.