Suppose that $\Gamma = (V\Gamma, E\Gamma)$ is a locally finite simplicial graph, i.e. $V\Gamma$ is a set and the set of edges $E\Gamma \subseteq \binom{V\Gamma}{2}$ consists of unordered tuples of vertices and that for every vertex $v \in V\Gamma$ the neighbourhood $N(v) = \{u \in V\Gamma\ \mid \{u,v\} \in E\Gamma \}$ is finite. Also, assume that $\Gamma$ is vertex transitive, meaning that for every pair of vertices $u,v \in V\Gamma$ there is a graph automorphism $\alpha \in \mathop{Aut}(\Gamma)$ such that $\alpha(v) = u$. This implies that all vertices are of the same degree $d \in \mathbb{N}$.
My question is following: is there a colouring of the edges of $\Gamma$ with $d+1$ colours? In the case when $V\Gamma$ is finite the answer is positive by Vizing's Theorem, but I am not sure whether the statement still holds if the the graph is infinititely countably many vertices.
Yes; since every finite subgraph of $\Gamma$ is $(d+1)$-edge-colorable, so is all of $\Gamma$ by the de Bruijn–Erdős theorem (applied to the line graph of $\Gamma$).