How to prove that the Möbius band has geodesics?

Normality is not dependent on orientation.

In $\mathbb R^3$, if I give you a plane $P$ and a vector $V$ based at a point of $P$, I can tell you whether $V$ is normal to $P$ without mentioning any orientation of $P$: $V$ is normal to $P$ if and only if $V \cdot W = 0$ for all vectors $W$ parallel to $P$. All I've used to formulate this definition is the (standard) inner product on the vector space $\mathbb R^3$.

You can now apply this principle at the point $\gamma(t)$, using the tangent plane $P = T_{\gamma(t)} S$ and the vector $V = \gamma''(t)$.