Answering this question would either require refuting choiceless large cardinals or getting close to refuting Woodin's HOD Conjecture.

First, if choiceless cardinals are consistent, one cannot rule out the existence of a second-order elementary embedding. Corollary 4.8 in Usuba's ["A note on Lowenheim-Skolem cardinals"][1] states that if there is a proper class of Lowenheim-Skolem cardinals, one can force the Axiom of Choice by a homogeneous definable class forcing. (He doesn't say homogeneous, but it will be.) Therefore assume there is a $j : V\to V$ and there is a proper class of Lowenheim-Skolem cardinals. Let $\mathbb P$ be Usuba's forcing, and let $G\subseteq \mathbb P$ be a generic filter. Then $j\restriction \text{Ord}$ is second-order elementary in $V[G]$: if $\varphi(\vec v)$ is a second-order  formula and $V[G]\vDash \varphi(\vec \alpha)$, then by homogeneity, $1_\mathbb P\Vdash \varphi(\vec \alpha)$, and so by the definability of $\mathbb P$ and the elementarity of $j$, $1_\mathbb P\Vdash \varphi(j(\vec\alpha))$, and hence $V[G]\vDash \varphi(j(\vec\alpha))$.

Second, assuming the HOD Conjecture, one can rule out the existence of a nontrivial second-order elementary embedding of the ordinals with critical point above the least extendible cardinal. This follows from Corollary 25 of Woodin-Davis-Rodríguez's paper ["The HOD Dichotomy."][2] The reason is that any second-order elementary embedding $j : \text{Ord}\to\text{Ord}$ extends to an elementary embedding from $\text{HOD}$ to $\text{HOD}$. It is also a prominent open question whether such an elementary embedding can exist, even assuming strong versions of the HOD Conjecture.

  [1]: https://arxiv.org/abs/2004.01515
  [2]: https://arxiv.org/abs/1605.00613