Here is one limit on the curvature needed. It is not optimal, but it is explicit, and probably a good starting point for numerical or theoretical optimization.
Proposition: Consider the vertices with their convex hull and its sides; let $h$ be the minimum over all sides of $$\frac{s_j}{\tan\left(\frac\pi4+\frac{t_j}2\right) +\tan\left(\frac\pi4+\frac{t’_j}2\right)}$$ where $s_j$ is the length of the side and $t_j$, $t’_j$ are half the angles at its vertices. Then there is a path through the vertices whose maximal curvature is $$\kappa_{\max} = \frac{1-\sin(\min t_i)}{h}$$ where again each $t_i$ is half of a vertex angle.
Proof: For consecutice vertices $UVW$, take a point $O$ at distance $h/(1-\sin t)$ from $V$ along the bisector of angle $UVW$. We will construct a path which includes the circle arc which is centered at $O$, goes through $V$, and is bounded by the two points $P$ and $Q$ with angular distance of $\frac\pi2-t$ from $V$.
For these points, $OP \perp UV$, $O$ is at distance $(h \sin t)/(1-\sin t)$ from $UV$, $P$ is at distance $h$ from $UV$ and the arc covers a distance of $(h \cos t)/(1-\sin t)$ or $h\tan(\frac\pi4+\frac{t}2)$ along $UV$. Similar claims hold for $OQ$ and $VW$.
Now we construct the path as the combination of all the arcs $PVQ$ and any needed lines connecting them, where all the lines would be at distance $h$ from the convex hull. The definition of $h$ ensures that none of the circle arcs overlap. Its curvature in each arc is $(1-\sin t)/h$, and its curvature in each line is $0$, so the maximal curvature is as stated. $\square$
Example: Let the vertices be $(\pm2,\pm1)$ and $(\pm3,0)$. Then each side length is $\sqrt{2}$ or $4$, and each vertex half-angle is $\pi/4$ or $3\pi/8$. So $h$ is the minimum of $$\frac{\sqrt{2}}{\tan\frac{3\pi}8+\tan\frac{7\pi}{16}}\sim0.19$$ $$\frac{4}{2\tan\frac{7\pi}{16}}\sim0.40$$ and $h\sim 0.19$, with a maximal curvature of $(1-\sin\frac\pi4)/h\sim 1.54$. The apparent minimum possible maximum curvature is $1$, achieved by the union of two semicircles and two horizontal lines.