Here's a conjectural solution, motivated by the idea of a "line shelling". Begin by choosing a "generic" nonzero vector $c$ in your ambient space, i.e. a vector such that it spans a line which crosses all of your hyperplanes.
Next, associate each vertex $v$ in the hyperplane arrangement to the unique region $R$ having the property that the dot product $c\cdot x$ is maximized on $R$ at $v$. Now assign to each vertex $v$ the set of hyperplanes which bound this region $R$ and pass through $v$. Each such $R$ is then bijectively mapped to the increasing chain in the intersection lattice poset labeled by this list of chosen hyperplanes arranged in ascending order.
Similarly, label each of the remaining regions $R$ by its set of bounding hyperplanes $\mathcal{H}$ which have the property that for each chosen $\mathcal{H}$ there is another bounding hyperplane $\mathcal{H'}$ of $R$ and a line segment going in the direction $c$ which starts at a point $p'$ on $\mathcal{H'}$ and ends at a point $p$ on $\mathcal{H}$, with $c\cdot x$ increasing as we progress along the segment from $p'$ to $p$. Bijectively map each of these regions $R$ to the increasing chain in the intersection poset labeled by its chosen collection of hyperplanes, again listed in increasing order.
It would be great if you or someone else wants to figure out whether this conjecture is correct-- maybe this particular approach to your question requires some hypotheses on the ordering of the hyperplanes? . I also wouldn't be surprised though if someone has thought about your question before -- your question certainly ties in with a lot of interesting work in the literature.
