It is true under the additional hypothesis that the locally finite graph is connected. Here the key observation is that the stabilizer of each vertex is compact. For a proof of this, see e.g. Lemma 1 in this paper.
However, local compactness does not hold in general. For instance, if the vertex set is countably infinite and there are no edges, then the automorphism group is just the symmetric group on a countable set and one can show directly from the definition of the compact open topology that the identity does not admit a compact neighborhood.
More generally, local compactness holds if there are finitely many connected components. Perhaps the necessary and sufficient condition is that there be only finitely many connected components $X_i$ such that $\operatorname{Aut} X_i$ is noncompact?