Here's a possible approach that goes backwards, where we define the tubular neighborhood first and the normal bundle second.
Let $M$ be a Finsler manifold and $S \subset M$ a smooth submanifold. Given any $x \in M$, we can define the distance $d(x,S)$ from $x$ to $S$ to be the shortest length of curves from $x$ to $S$. We'll call a curve segment $S$-minimizing if one endpoint lies in $S$ and the length of the curve equals the distance from the other endpoint to $S$.
Then one can define a tubular neighborhood of $S$ of radius $r$ to be the set of all possible endpoints of $S$-minimizing geodesics. However, this is simply the set of all points within distance $r$ from $S$.
We can now define a subset $N_*S \subset T_*M$ with respect to the Finsler metric as follows: $v \in T_pM$ lies in $N_pS$ if there exists an $S$-minimizing curve starting at $p$ such that $v$ is tangent to the curve at $p$.
Using the existence and uniqueness of a geodesic with given starting point and velocity, we can define the exponential map $e: N_*S \rightarrow M$.. Moreover, this should show that $N_*S$ is in fact a vector bundle over $S$ and the exponential map defines a diffeomorphism from a neighborhood of the zero section of $N_*S$ onto a neighborhood of $S$ in $M$. In particular, given any precompact subset of $S$, there exists $r>0$ such that the exponential map is a diffeomorphism onto the tubular neighborhood of radius $r$ from its preimage.