Let $X$ be a smooth vector field on a manifold $M$. The Jacobian of $X$ at a critical point $x^*$ is the linear map $$X'(x^*): T_{x^*}M \rightarrow T_{x^*}M$$ where $T_{x^*}M$ is the tangent space to $M$ at $x^*$, defined by $$X'(x^*) \cdot u = \frac{d}{dt} \left( \text{d}_{x^*}\Theta_t \cdot u \right)|_{t=0}$$ for any $u \in T_{x^*}M$.
Here $\Theta_t: M \rightarrow M$ is the flow of $X$ and $\text{d}_{x^*}$ is the differential, so $\text{d}_{x^*}\Theta_t$ is a linear map of $T_{x^*}$ to itself, since $x^*$ is a critical point: $$\text{d}_{x^*}\Theta_t: T_{x^*}M \rightarrow T_{\Theta_t(x^*)}M = T_{x^*}M$$
Chosen a local chart at $x^*$, the matrix representation of this linear map is the usual Jacobian matrix: $$\left[ X'(x^*) \right]_{ij} = \left( \frac{\partial X^i}{\partial x^j} \right)_{x = x^*}$$
See e.g. Abraham-Marsden (1978), Foundations of mechanics, p. 72 (page attached below).
Question Is there an analogue coordinates-independent definition for the "Jacobian" of a 1-form $\alpha$ on a manifold $M$ as a linear map on a cotangent space and such that the local representative is the usual Jacobian matrix?
The answer is yes, and, in fact, it is a more general notion. The construction is often called the intrinsic derivative. Any time you have a smooth section $s$ of a vector bundle $E\to M$, its derivative at a zero $p$ is a well-defined linear map $Ds_p\colon T_pM\to E_p$.
Of course, you can work in a trivialization on an open set $U$ containing $p$ and then $s$ corresponds to a function $f_U\colon U\to\Bbb R^k$ for $V\subset\Bbb R^n$ and we can consider the derivative $(Df_U)_p\colon T_pM \to \Bbb R^k$. If you have an additional, overlapping trivialization on $V$, then on $U\cap V$ we have $f_V(x)=g_{UV}(x)f_U(x)$, where $g_{UV}\colon U\cap V\to GL(k)$ is the transition function of the vector bundle. In general, $(Df_V)_p = g_{UV}(p)(Df_U)_p + (Dg_{UV})_p f_U(p)$, but at a zero of the section, the second term vanishes and we see that $(Df_U)_p$ defines a well-defined mapping $T_pM\to E_p$.