Automorphism groups and etale topological stacks

I don't think this is true. Let $X$ be the quotient of the action of $\mathbb Q$ on $\mathbb R$ by translation. This is a sheaf, and its automorphism groups are trivial. Suppose that there exist a local homeomorphism $U \to X$, where $U$ is non-empty a topological space. Let $V \to U$ be the pullback to $U$ of the $\mathbb Q$-torsor $\mathbb R \to X$; then $V\to \mathbb R$ is a local homeomorphism. By restricting $U$ we may assume that $V = U \times \mathbb Q$; but then $V$ can't be locally connected, and this is a contradiction.

Here's a counterexample: the stack associated to the relative pair groupoid of the map $$ ([0,1]\times\{0\}) \cup (\{1\}\times[0,1]) \cup ([1,2]\times\{1\})\;\;\to\; [0,2] \qquad\qquad\qquad\qquad\qquad $$

$$\qquad\qquad\qquad(x,y)\;\;\mapsto\;\;\; x$$

Equivalently, this stack can be described as the pushout in the 2-category of stacks of the diagram $[0,1]\leftarrow \{1\} \rightarrow [1,2]$ (where we identify a space with the stack it represents).