Preliminaries
The exponential map $\exp : TM \rightarrow M$ is defined by $\exp{(v)} = \gamma_v (1) $ where $\gamma_v$ denotes the geodesic starting at $p \in M$ and initial velocity $v \in T_p M$. The restricted exponential map $\left. \exp \right|_{T_p M}$ is denoted $\exp_p$.
Setup
Let $(M, g)$ be a smooth, compact Riemannian n-manifold without boundary. Let $S \subset M$ denote a smooth submanifold of codimension-0 with boundary. Fix $X \in \mathfrak{X}(M)$.
Question
Let $T_X : M \rightarrow M$ be defined by $$ T_X (p) := \exp{(X(p))} $$
Are there any conditions on $X$ such that the restriction $\left. T_X \right|_{S} $ becomes a diffeomorphism onto its image?
Notes
I am aware that the exponential mapping restricted to $T_p M$ is a diffeomorphism on some neighborhood of the origin. I would guess that we would need to specify that $X$ belong to the intersection of all the neighborhoods where where $\exp$ is a diffeomorphism for each $p \in M$. But I am not sure.
Differential in terms of Jacobi fields
So to answer @Artic Char in the comments I will do the computation by applying the chain rule. Disclaimer: I'm not sure if the computations below are correct.
The differential of the exponential map at $(p, v) \in T M$ can be expressed in terms of Jacobi fields. We can write $$ \begin{aligned} d_{(p, v)} (\exp) : T_{(p, v)}(T M) &\rightarrow T_{\exp_p(v)} M\\ (\hat{p}, \hat{v}) &\mapsto J(1) \end{aligned} $$ where $J$ is the Jacobi field along the geodesic $\gamma(t) = \exp_p(tv)$, satisfying initial conditions $J(0) = \hat{p}$ and $D_t J(0) = \hat{v}$.
From another side, we can work in natural coordinates around $p$ so that we make the identification $T_{(p, v)}(T M)$ with $\mathbb{R}^n \times \mathbb{R}^n$ and similarly with $T_pM$ and $\mathbb{R}^n$. The differential of a vector field is then given by $ d_p X(v) = (v, \Lambda v)$ where $\Lambda$ is a linear operator $\mathbb{R}^n \rightarrow \mathbb{R}^n$. Hence by the chain rule we have that
$$ \begin{aligned} d_p T_X(v) = J(1) \end{aligned} $$ where J is the solution to the Jacobi equation (in the given coordinates) along the geodesic $\gamma_{X(p)}$ with initial conditions $J(0) = v$, $D_t J(0) = \Lambda v$.
A simple sufficient condition is that $X=0$. Then the map $T_X$ you defined is the identity map, hence, its restriction to any submanifold is a diffeomorphism to the image.