"Stoning" is not mentioned on the "Oedipus Rex" Wiki page, but it is cited on the "Stoning" Wiki page. It says in the article:
Stoning is also mentioned in Ancient Greek mythology.
The note at the bottom reads:
Oedipus asks to be stoned to death when he learns that he killed his father.
"Stoning" is not mentioned at the end of Oedipus Rex, but it is mentioned in the middle of Oedipus at Colonus. Oedipus is speaking to Ismene, his daughter:
Then may the gods ne'er quench their fatal feud, And mine be the arbitrament of the fight, For which they now are arming, spear to spear; That neither he who holds the scepter now May keep this throne, nor he who fled the realm Return again. They never raised a hand, When I their sire was thrust from hearth and home, When I was banned and banished, what recked they? Say you 'twas done at my desire, a grace Which the state, yielding to my wish, allowed? Not so; for, mark you, on that very day. When in the tempest of my soul I craved Death, even death by stoning, none appeared...
As it was written by the same author (Sophocles), it can be concluded that Oedipus does, in fact, contemplate death by stoning, but I don't see it as overt or in connection with his father's death. Rather, it is a general and reflective statement, almost as an afterthought (as it is surrounded by commas), about death. The fact that Oedipus mentions a form of capital punishment (stoning) as opposed to suicide is important. Oedipus is a tragic hero because, as the previous editor says, he takes responsibility for his crimes and refuses to escape them by death, by suicide or public execution.
Not to disparage what is found online, but I guess I am not really sure that such a finding is accurate. I don't think one can find Oedipus asking to be stoned. Part of what makes the ending so powerful is that Oedipus inflicts all of the punishment that he needs to impose upon himself. He does not need society to do anything more. The ending in which he pleads for understanding for his children and then takes it upon himself to be exiled is all he requires. I don't see anything of stoning or any sort of public form of punishment being asked. Oedipus assumes the role of adjudicator himself and does not require anyone else. In the absence of a structure or order, Oedipus assumes it himself to right what was wrong and bring some semblance of order back to both his life and to Thebes. I don't see anything relating to public punishment in this conception.
In Oedipus Rex, the protagonist just asks for exile, the curse that he put on himself. He does not mention "stoning" himself, therefore, I would say no to that question.