Let $X$ be a scheme with a smooth surjective morphism onto a stack $X\to \cal X$. Why does it admit local sections?
In Why a smooth surjective morphism of schemes admits a section etale-locally? is explained how to prove this fact if $\cal X$ is also a scheme, but I don't see how to generalise to stacks this result.
I will follow the notation in the OP's comment on the question.
Given $x:S \to \mathcal X$, pull-back the given smooth surjective morphism $X \to \mathcal X$ to obtain a smooth morphism $T \to S$.
I am going to assume that $\mathcal X$ is an algebraic stack; then (by definition) its diagonal is representable by an algebraic space, and so, since $S$ is a scheme, the pull-back $T$ is an algebraic space. Thus we can find a surjective etale morphism $U \to T$ with $U$ a scheme.
The composite morphism $U\to T \to S$ is then smooth and surjective, and the composite morpism $U \to T \to X$ lifts $x$. We have thus made a smooth-local lift of $x$.
Now any smooth morphism $U \to S$ of schemes admits an etale local section, i.e. we can find $V \to S$ etale and surjective which lifts to a morphism $V \to U.$ The composite $V \to U \to X$ is then an etale local lift of $x$, as required.