There is a set theoretic axiom due to Paul Corazza called the Wholeness Axiom, which is stated in the language of ZFC augmented by a single unary function symbol $j$. The axiom expresses, as a scheme, that $j$ is a nontrivial elementary embedding from $V$ to $V$. That is, we have the elementary axiom scheme, expressing "for all $x$, $\phi(x)$ iff $\phi(j(x))$" and the nontriviality axiom, expressing "exists $x$, $j(x)\neq x$" and the critical point axiom, expressing "there is a least ordinal $\kappa$ such that $\kappa < j(\kappa)$".
Under this axiom, $j$ really is an elementary embedding from the universe $V$ to $V$, and so this presumably induces the kind of functor you want.
The point is that the large cardinal consistency strength of this axiom is weaker than a Reinhardt cardinal. In fact, it is strictly below an I3 cardinal.
But the situation with this axiom is not great, since the $j$ you get will not be a definable class in the usual sense of ZF. Also, you will not have the Replacement Axiom in the full language with $j$. So to make use of the axiom, you in effect give up a little of what you mean by the existence of such a $j$.
There is an ambiguity, isn't there, in the question when you ask about the existence of a proper class object. What kind of existence is desired? The question is not directly formalizable in ordinary set theory, since the question is itself a quantification over proper classes (although Kelly Morse set theory would accommodate this). Do you want a definable class? Do you want a class in the sense of Goedel-Bernays? Having a relaxed attitude about this allows the large cardinal consistency strength of the answer to come down.
