Assume we are given an embedded Riemannian submanifold $(\mathcal{M},g)\subset (\overline{\mathcal{M}},\overline{g})$, with $\overline{\mathcal{M}}$ having the Levi-Civita connection $\nabla$ compatible with its Riemannian metric $\overline{g}$. $\mathcal{M}$'s Levi-Civita connection $\nabla^\mathcal{M}$ relates to $\nabla$ via orthogonal projection. Now, usually the definition of $\nabla^\mathcal{M}$ is that it is a bilinear operator on the space of vector bundles on $\mathcal{M}$, so sections of $\mathcal{M}$'s tangent bundle $T\mathcal{M}$.
Now I wonder, why we cannot apply $\nabla^\mathcal{M}$ to sections of $\mathcal{M}$'s normal bundle $N\mathcal{M}$? In my mind, it makes sense to talk about covariant derivatives of normal vector fields $N(t)$ along curves $c:I\rightarrow \mathcal{M}$ (i.e. $\nabla^\mathcal{M}_{c'(t)} N(t)$), since the tangent space $T\overline{\mathcal{M}}\big|_\mathcal{M}$ splits as $\bigsqcup _{x\in \mathcal{M}} T_x\mathcal{M} \oplus N_x\mathcal{M}$.
Actually, you can. Have a look at Proposition 4.26 in Lee's Introduction to Riemannian Manifolds.