Dear {Recipient},
I would like to apologize on behalf of my child, {Name}. Her behavior at {occasion/place/date} was unacceptable. In our family, we certainly do not consider {bad behavior} appropriate in any way. {Name} has been punished with{punishment}, and while she is too young to compose a letter of her own, rest assured that she is very sorry for her behavior.
I genuinely hope that this incident will not affect our friendship or the friendship of our children. Again, on behalf of our family and {Name}, please accept my most sincere apologies.
Yours Truly,
{Sender}