This seems to be a true statement, but I am not sure if I can write down a valid proof:
Let $\Omega \subseteq \mathbb R^2$ be a bounded simply connected open set (with no regularity assumptions on $\partial \Omega$). Then for all $\epsilon > 0$, we can find a compact $K_\epsilon \subseteq_c \Omega$ with $\partial K_\epsilon$ being $C^1$, such that $d(x,\partial \Omega) < \epsilon$ for all $x \in \partial K_\epsilon$.
My first thought was to use smooth approximation of continuous functions, provided by tubular neighborhood theorem. But since there is no regularity assumptions on $\partial \Omega$, it may not even be a Jordan curve, so I would not be able to (even locally) consider $\partial \Omega$ as the graph of a continuous function.
As specified, the task is unexpectedly easy: Pick $a\in\partial \Omega$. By definition of boundary, the $\epsilon$-ball around $a$, $B_\epsilon(a)$, intersects $\Omega$. Pick $b\in\Omega\cap B_\epsilon(a)$. Then $B_r(b)\subseteq_c \Omega$ provided $r$ is small enough, say $r<r_0$. Now let $$K_\epsilon = B_{\min\{r_0,\epsilon-d(a,b)\}/2}(b)$$ Its boundary is a circle and each of its points is less than $\epsilon$ away from $a\in\partial\Omega$.