Let $M$ be a complete riemannian manifold, $L$ a smooth submanifold of $M$ and $\gamma$ a geodesic with $\gamma'(0)$ normal to $L$. A focal point of $L$ is a critical value of the normal exponential $\exp:\nu L\to M$. So the question is: is the set of focal points $\gamma(t)$ of $L$ discrete?
I know that's related to the Morse index theorem, but all the versions of it I've found are too general and do not contain a conclusion about finiteness.
No, not necessarily. Let $M$ be the sphere $S^2$. Let $L_0$ be the equator and $\gamma$ some meridian going to the North pole $N$ (which is a focal point of $L_0$). Pick a sequence of distinct points $z_n$ on $\gamma$ converging to $N$, and a sequence of distinct points $x_k$ on the equator converging to $\gamma(0)$.
For each $n$, smoothly perturb $L_0$ near $x_n$ so that the perturbed curve includes a short subarc of the circle $\{x: d(x,z_n)=d(x_n,z_n)$. This will make $z_n$ a focal point of the perturbed equator $L$. The intervals on which perturbation is made can be kept disjoint, since the arcs to be included in $L$ can be arbitrarily short. The perturbations accumulate at $\gamma(0)$, so one has to be careful to make sure that $L$ is smooth ($C^\infty$) there. But this can be done because $z_n$ can converge to $N$ arbitrarily fast, making perturbations as small as we wish, in every $C^k$ norm.