Is the failure to document sources in a research paper considered theft, and what are the conditions for repentance if that is the case, especially given that the supervisor has asked to forget the matter?

There is no objection to quoting, provided two conditions are met: fidelity in transmission and attributing the quoted material to its author. Failure to do so may cause the person to fall under the category of "one who pretends to have been given what he has not been given" and "whoever cheats is not of us" if it results in cheating. However, if it was not intentional, there is no sin upon him, for Allah Almighty says: "And there is no blame upon you for that in which you have erred, but [only for] what your hearts intended," and because the Prophet, peace and blessings be upon him, said: "Indeed, Allah has pardoned for me my Ummah their mistakes, their forgetfulness, and what they are coerced into."