Ackermann found an explicit canonical bijection between $\omega$ and $V_{\omega}$.
Is it possible to somewhat explicitly define such bijections between $\kappa$ and $V_{\kappa}$ for an inaccessible cardinal $\kappa$ (possibly with extending or modifying Ackermann's idea)?
Off the bat, the answer is negative. It of course consistent that $\kappa$ is strongly inaccessible (in the strong sense of the term), and $\sf AC$ fails in $V_\kappa$, so there is no bijection between $\kappa$ and $V_\kappa$.
This tells you that some use of choice is necessary, and that it cannot be "too explicit". But more than just that. The Ackermann map is internal to $V_\omega$, and any such "explicit map", even if relying on some choice somehow, would end up being internal to $V_\kappa$. So that means that $V_\kappa$ is not just a model of $\sf ZFC$, but also of global choice. However, we know that this is not always the case. For example, if we add two infinite sets of Cohen subsets to every regular cardinal below $\kappa$, then there is no definable linear ordering of $V_\kappa$ (where definable is "internally definable", of course).
What you need in order to have this sort of map is some sense of being able to stratify $V_\kappa$ in a way that lets you well-order each of the levels uniformly. So, for example, in $L$, this would be the case with $<_L$. But in general? I don't see a reason why any such function exists.