As is often the case with these quotes, it's actually a summary of a summary of Plato.
We see an early version in Ernst Cassirer's 1944 essay An Essasy on Man:
It is impossible—says Plato in the Republic—to implant truth in the soul of a man as it is to give the power of seeing to a man born blind...Here we have the new, indirect answer to the question "What is man?" Man is declared to be that creature who is constantly in search of himself—a creature who in every moment of his existence must examine and scrutinize the conditions of his existence.
(That footnote is to Cassirer's quote above.)
He then goes on to quote that which we all know, that "the unexamined life is not worth living."
Man being a creature in search of himself is a rather old saying, and I've yet to come across anyone saying that Plato said it directly. Examples abound: 1, 2, 3.
It doesn't quite end there, though. In his 1954 monograph The Insecurity of Freedom, Abraham Joshua Heschel quotes Cassirer, but introduces a further summary:
Nowhere in Plato’s Socratic dialogues do we find a direct solution to the problem, "What is man?" There is only an indirect answer, "Man is declared to be that creature who is constantly in search of himself—a creature who in every moment of his existence must examine and scrutinize the conditions of his existence."18 He is a being in search of meaning.
Notice how it starts off by saying there is no direct answer, and then his final sentence is outside of quotes.
From these, I'd say this is not something Plato wrote but is instead a close summary of Heschel's expanded summary of a summary by Cassirer.