Is immediate initiative or procrastination obligatory when giving charity to redress injustices inflicted upon unknown individuals prior to puberty, given current financial capability, and does delaying this lead to sin?

Childhood does not prevent financial obligations from attaching to a minor's responsibility. If he damages something, he is liable for its value, and he is obligated to compensate for theft from his own wealth. Restitution for injustices must be made immediately when able, because repentance is obligatory immediately, as is the return of injustices. If the owner of the injustice is unknown, one is absolved by giving the money in charity on their behalf, or by increasing good deeds and supplicating to Allah.