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 $v$.)