Here is a cool proof that if there is a nonvanishing vector field then the Euler characteristic must be zero:
Suppose $X$ is a nonvanishing vector field. Let $f_t$ be the corresponding time $t$ flow. Then for some small $\varepsilon > 0$, the flow $f_\varepsilon$ has no fixed points (to show this, we must use the compactness of the manifold $M$). So by the Lefschetz fixed point theorem, we have $$0 = \sum_i (-1)^i \operatorname{Tr}(f_{\varepsilon,\ast} : H_\ast(M) \to H_\ast(M)). $$$$0 = \sum_i (-1)^i \operatorname{Tr}(f_{\varepsilon,\ast} : H_i(M) \to H_i(M)). $$ But since $f_\varepsilon$ is homotopic to $f_0 = \operatorname{Id}$, we have that the RHS is equal to $$\sum_i (-1)^i \operatorname{Tr}(\operatorname{Id}:H_\ast(M) \to H_\ast(M)) = \sum_i (-1)^i \operatorname{rk}H_\ast(M),$$$$\sum_i (-1)^i \operatorname{Tr}(\operatorname{Id}:H_i(M) \to H_i(M)) = \sum_i (-1)^i \operatorname{rk}H_i(M),$$ which is the Euler characteristic.