Indeed, $\ker\nabla$ is a well-defined subsheaf of $\mathcal{E}$. In particular, it is a sheaf of finite-dimensional vector spaces with the stalk-rank $\dim(\ker\nabla)_x$ bounded by the vector bundle rank $\text{rk}$ $E$. But $\ker\nabla$ is not a local system any more if $\nabla$ is not flat. That happens if and only if
$$
x\mapsto \dim(\ker\nabla)_x
$$
is a locally constant function on $X$.

As far as I know, in the smooth setting, the best we can get is the following: define
$$
X^{\leq d}:=\{x\in X | \dim(\ker\nabla)_x\leq d\}\,.
$$
Then $\{X^{\leq d}\}_d$ is a locally finite collection of closed sets in $X$, and the restriction of $\ker\nabla$ to the subsets $X^{\leq d}-X^{\leq d-1}$ is locally constant for all $d$ ($\leq \text{rk}$ $E$).

This is proved in Brian Conrad's notes on the Riemann-Hilbert correspondence (link). Maybe there's a better/different description of such sheaves. **Edit:** They're clearly *constructible sheaves* on $X$ (equivalently, representations of the exit-path category (see Treumann)).

This is a result that relies on *uniqueness* of local parallel sections (as a consequence of the initial value problem for first-order ODEs), but tells nothing about existence. In fact, the sheaf $\ker\nabla$ may as well be empty, i.e. concentrated in $X^{\leq 0}$: an element of the fiber $\mathcal{E}_x$ may not extend even locally to a parallel section. The obstruction to the *existence* of such solution lies exactly on the curvature of the connection. (When the connection is flat, parallel transport is independent of path, and we may use parallel transport of a vector $v\in \mathcal{E}_x$ in a small simply connected neighborhood of $x$ to define a parallel section of $\mathcal{E}$ extending $v$.)