Is it permissible to exact retribution for injuries without referring to the ruler (wali al-amr), such as by injuring the aggressor in the same way he assaulted [the victim]?

It is never permissible to take one's right by force. Retaliation (Qisas) is not legislated for an injury unless the wound reaches a bone, otherwise, there is no Qisas due to the impossibility of achieving exact equivalence. Furthermore, it is a condition for executing Qisas that the ruler or his deputy be present, lest injustice occur due to a desire for revenge.