Homology and cohomology of 7-manifold

For any connected manifold $M$, there is a homomorphism $\pi_1(M)\to\mathbb{Z}/2$ which sends a loop to $0$ if going around the loop preserves orientation and sends the loop to $1$ if going around the loop reverses orientation. This homomorphism is trivial iff $M$ is orientable. Since $\mathbb{Z}/2$ is abelian, this homomorphism factors through the Hurewicz map $\pi_1(M)\to H_1(M)$. In particular, this means that if $H_1(M)=0$, the homomorphism is trivial so $M$ is orientable.

(By the way, the statement that $\chi(M)=0$ does not require $M$ to be orientable--you can prove it using mod $2$ Poincare duality, for instance.)

A quick proof using Stiefel–Whitney classes: a manifold $M$ is orientable iff the first SW class $w_1(M) \in H^1(M;\mathbb{Z}/2\mathbb{Z})$ is zero. But by the universal coefficient theorem, $$H^1(M;\mathbb{Z}/2\mathbb{Z}) = \operatorname{Hom}_\mathbb{Z}(H_1(M;\mathbb{Z}), \mathbb{Z}/2\mathbb{Z}) = 0.$$

Of course under the hood I don't think there's anything more than what you can find in Eric Wofsey's answer, but this argument is quite simple and shows the power of characteristic classes.