This word came into English from Latin at the end of the 14th century referring to any kind of wrongful act. Most of Shakespeare's usage reflects this breadth, keeping well away from the modern sense of physical injury.
When Worcester talks about "the injuries of a wanton time" (Henry IV Part 1, V.i.50), he means "wrongs" or "grievances". When Oberon tells Titania "Thou shalt not from this grove Till I torment thee for this injury" (A Midsummer Night's Dream, II.i.147) he means "insult" or "slight". There is no suggestion that Titania has physically hurt him.
The nearest we get to the modern sense is when Montjoy reports the words of the French king to Henry V (Henry V, III.vi.120): "we thought not good to bruise an injury till it were full ripe." This might mean simply "hit back at a wrong", but here injury is more likely to mean a "sore" or "abscess".
The Second Quarto of Romeo and Juliet also has the word as a verb: "I never injuried thee", says Romeo to Tybalt (III.i.67). If it is a genuine usage it must mean "do (you) an injustice".