I assume you meant for the fiberwise rank of $E$ to be constant, say $r>0$ (or at least uniformly bounded above). The answer is "yes" when the Stein space has finite dimension (equivalently, when its analytic irreducible components, all of which must be equidimensional via consideration of their connected normalizations, have uniformly bounded dimension). There is no need to make a "global finite generation" hypothesis (which would be hard to check in practice anyway); dimension finiteness of $X$ is entirely sufficient.

In fact, for induction purposes it is better to consider more generally $E$ that is merely a coherent sheaf for which its fibers $E(x)$ at all $x \in X$ have uniformly bounded dimension, say at most some $r>0$. We will show that any such $E$ is generated by finitely many global sections. Then when $E$ is a vector bundle with fiberwise rank $\le r$ we can build a finite trivializing cover governed by where various subsets of size $\le r$ in the global generating set constitute a fiberwise frame. The main content in the argument will be the global theory of analytic irreducible components and their equidimensionality.

We may assume $X$ is non-empty. Let $d \ge 0$ be the dimension of $X$. If $d=0$ then $X$ is topologically discrete (with artinian local ring at each point) and everything is clear. Suppose $d > 0$, and let $\{X_i\}$ be the (locally finite) set of analytic irreducible components of $X$. For each $i$, choose $x_i \in X_i$ not lying in any other $X_j$ (as clearly exists by local finiteness of $\{X_i\}$ in $X$ or many other reasons). Let $\{e_j^{(i)}\}_{1 \le j \le r}$ be a spanning set of the fiber $E(x_i)$.

The set $Z$ of points $\{x_i\}$ in $X$ is discrete, and with the reduced structure we get a closed immersion $h:Z \hookrightarrow X$. Let $s_1, \dots, s_r$ be global sections of $E$ such that $s_j(x_i) = e_j^{(i)}$; this exists because of the Stein property of $X$ and the surjectivity of the map of coherent sheaves
$$E \rightarrow h_{\ast}(E|_Z) = \bigoplus_i (h_i)_{\ast}(E(x_i))$$
where $h_i: \{x_i\} \hookrightarrow X$ is the natural closed immersion
(surjectivity uses the discreteness of $Z$).

Now consider the natural map $\phi:O_X^r \rightarrow E$ defined by $(a_j) \mapsto \sum a_j s_j$. By design, $\phi$ is surjective between fibers at all points of $Z$, so the coherent sheaf $F = {\rm{coker}}(\phi)$ on $X$ has vanishing stalk at each $z \in Z$. Hence, the coherent ideal sheaf ${\rm{Ann}}_{O_X}(F)$ has associated closed subspace $X' \subset X$ whose intersection with each irreducible $X_i$ is a proper analytic subspace (as $x_i \not\in X'$ for all $i$). Since each irreducible component of $X'$ is contained in one of $X$ (by local finiteness considerations with analytic sets) it is therefore clear that the dimension $d'$ of $X'$ is strictly smaller than $d$. (If we are so lucky that $X'$ is empty then I suppose we declare $d' = -\infty$; whatever.)

By induction on dimension, $F$ viewed as a coherent sheaf on $X'$ (with all fibers of dimension at most $r$) is generated by a finite set of global sections. These lift to global sections of $E$ due to the Stein property of $X$, and together with the $s_j$'s above constitute a finite global generating set of $E$ due to Nakayama's Lemma.

QED