Suppose we have a contact submanifold $i:(S,\xi_S) \hookrightarrow (M,\xi)$. This means that if $\xi = \ker \alpha$, then $\xi_S = \ker i^* \alpha$.
What is the Reeb vector field $X_S$ on $S$ and how does it relate to the Reeb vector field $X_M$ of $M$? I thought we might try studying a tubular neighborhood of $S$ viewed as the normal bundle: $\pi:N_S \to S$ and $i:S \hookrightarrow N_S$ is the zero section. Then $X_M \in TN_S = TS \oplus V$ where $V$ are the vertical fibers; we can project $\pi_* X_M \in TS$ and I thought perhaps this should be $X_S$.
However, $i^*\alpha(\pi_*X_M) = (i \circ \pi)^* \alpha(X_M)$. Unfortunately, $\pi \circ i = \text{id}$ but $i \circ \pi \neq \text{id}$.
In general, I believe there exists no easy relation between the Reeb vector field of a contact submanifold and the Reeb vector field of the ambient contact manifold.
Example. Let $(M,g)$ be a Riemannian manifold, then $STM$, which is the unit tangent bundle of $(M,g)$, is canonically a co-oriented contact manifold whose Reeb flow is the geodesic flow on $M$. Similarly, if $N$ is a submanifold of $M$, then $STN$ is a contact submanifold of $STM$ whose Reeb flow is the geodesic flow on $N$. However, unless $N$ is totally geodesic, it is difficult to relate the geodesic flow on $N$ and the geodesic flow on $M$, so that it is illusionary to express the Reeb vector field of $STN$ in terms of the Reeb vector field of $STM$.