At What Age Is Hijab Mandatory?
Question

My question is: if a female child has begun to grow body hair, is she required by the Shari`ah to observe full hijab?
Praise be to Allah.The age of accountability in Islam
A person is not considered to be accountable until after reaching puberty. Before puberty he or she is not accountable. The Prophet (peace and blessings of Allah be upon him) said: “The Pen has been lifted from three: from the child until he reaches puberty, from the sleeper until he wakes up, and from the one who has lost his mind until he recovers.” (Narrated by Abu Dawud, 4402)
 Based on this, then a girl has to put on complete hijab when she reaches puberty.  
Signs of puberty for males and females in Islam
There are three signs of puberty that males and females have in common: 

Nocturnal emissions (wet dreams)
Growth of coarse hair around the private parts
Reaching the age of fifteen 

Sign of puberty for females in Islam
In the case of females, there is a fourth sign which is: 

Menstruation 

Importance of training children on doing obligatory duties 
If one of these signs of puberty appears in a girl, then she is obliged to do all the obligatory duties and to avoid all haram things. One of the obligatory duties is wearing hijab . 
But the parent or guardian of a girl has to make her get used to doing the obligatory duties and avoiding haram things before puberty, so that she will grow up with that and it will not be too hard for her to adhere to that after she reaches puberty. This is one of the basic principles of child rearing that are established in the Shari`ah. 
The Prophet (peace and blessings of Allah be upon him) said: “Train your children to pray when they are seven years old, and smack them (lightly, that is, discipline them) if they do not do so when they are ten, and separate them in their beds.” (Narrated by Abu Dawud, 495; Ahmad, 2/187, from the hadith of ‘Amr ibn Shu’ayb, narrated from his father from his grandfather.) 
Something similar was mentioned in the hadith of Saburah ibn Ma’bad which was narrated by Abu Dawud (494) and al-Tirmidhi (407). He said, it is hasan sahih. The hadith was classed as sahih by al-Albani in al-Irwa, 247. 
Al-Bukhari (1960) and Muslim (91136) narrated in their sahihs from al-Rubayyi’ bint Mu’awwidh the hadith concerning the fast of ‘Ashura and when it was enjoined on the Muslims. 
In this hadith, it says: “And after that we used to fast it – meaning ‘Ashura – and make our small children fast, and we would go to the mosque and make them toys out of wool, and if one of them cried for food we would give them that toy (to play with) until iftar.” 
According to a report narrated by Muslim: If they asked us for food, we would give them the toy to distract them until they completed their fast. 
Al-Nawawi said in Sharh Sahih Muslim (8/14): 
“This hadith describes training children to do acts of obedience and making them get used to doing acts of worship, but they were not accountable.” 
Ibn al-Qayyim said in Tuhfat al-Mawdud bi Ahkam al-Mawlud (p. 162): 
“Even though the child is not accountable, his guardian is accountable and it is not permissible for him to enable him to do haram things, so that he becomes used to them and it is difficult to wean him away from them afterwards.” 
And Allah knows best and is most wise. 
So if a girl is approaching puberty, there is the fear that her not wearing hijab may cause young men to be tempted by her or her by them. Hence in this situation, her parent or guardian has to make her wear hijab so as to prevent means that may lead to evil or immorality.
And Allah knows best.