Does metrizable, locally compact and $\sigma$-compact imply the separability of $C_{0}(M)$?

Let $M$ be a locally compact and metrizable space. If $M$ is $\sigma$-compact, prove $C_{0}(M)$ is separable.

In addition, discuss whether or not there is a natural way to weaken the hypotheses. For example, does local metrizability suffice?

I'm aware that, as a consequence of local compactness, for any $x \in M$ there is a $\epsilon>0$ such that $B(x;\epsilon)$ admits compact closure. In addition, since $M$ is $\sigma$ compact, each open cover of $M$ has a countable subcover.

But this amounts to little more than restating definitions. I can't seem to find an approach to solve this problem.