Suppose we have a non-compact $3$ dimensional manifold $M$ which is spinable. Fix a CW structure of the manifold, then being spinable is equivalent to the tangent bundle being trivial on the $2$-skeleton. There are two ways to proceed now. One can either use that every non-compact $n$-manifold is homotopy equivalent to a subset of its $n-1$-skeleton or one can use that $\pi_2(SO(3))$ vanishes and hence every trivialization on the two skeleton can be extended to the $3$-skeleton.
So all that is left to show is that an orientable non-compact $3$-manifold is spinable. Since $H^2(M;\mathbb{Z}/2)\cong \text{Hom}(H_2(M;\mathbb{Z}/2),\mathbb{Z}/2)$ the second Stiefel-Whitney class is non-zero if and only if there exists an element $\alpha$ in $H_2(M;\mathbb{Z}/2)$ such that $w_2(\alpha)\neq 0$. Note that $\alpha$ can be represented by an embedded compact surface $S$ i.e a subsurface such that $\alpha=[S]$ (Represent $\alpha$ by an immersed subsurface and then resolve the singularities). Since $S$ is compact we know that it is contained in a compact $3$-dimensional submanifold $N$. Since $i*(w_2(M))=w_2(N)=0$$i^*(w_2(M))=w_2(N)=0$ we see that this implies that $w_2(M)([S])=0$ and hence $w_2(M)$ is zero i.e. $M$ is spinable.
