"Completing" a vector field on a non-compact manifold $M$

Pick any complete Riemannian metric on your manifold (such things exist; see this) and normalize the vector field —you can do this because the vector field does not vanish anywhere.

Pick a point $p$ and let $(a,b)$ be the maximal domain of the integral curve $\gamma$ of the resulting vector field which starts at $p$ at time $0$. If $b<\infty$, then $\gamma(t)$ must leave every compact set as $t\to b$, so in particular it leaves every closed ball centered at $p$ as $t\to b$ (closed balls are metrically compact in a geodesically complete manifold: this is one form of the Hopf–Rinow theorem; see this for a statement) Since the curve is parametrized by arc length, this is impossible.

A similar argument shows that $a$ must be $-\infty$, so the normalized vector field is complete.