Current state of Straus's illumination problem

As far as I know, the specific question you ask (entirely illuminable from at least one point) remains open. (Tokarsky showed that placing a light in some spots can leave some points dark.) But you may be interested to know that an old conjecture has been settled, as I reported in this earlier question:

Lelièvre, Monteil and Weiss have shown that if $P$ is a rational polygon, for every $x$ there are at most finitely many $y$ not illuminated by $x$.

Lelievre, Samuel, Thierry Monteil, and Barak Weiss. "Everything is illuminated." arXiv:1407.2975 (2014).

This started as a comment, but became too long.

It might be worth mentioning that from a physics point of view, this problem has a certain ambiguity that somewhat diminishes its interest: when the walls are mirrors it is clear what happens when a light ray hits the wall (specular reflection), but what happens when a light ray hits a vertex? The answer to the "illumination problem" depends crucially on how one treats the vertices, because the "dark points" (points inside the region that cannot be illuminated by a point source of light) may appear only if one assumes that a light ray that hits a vertex is extinguished. (Diffuse reflection, rather than specular reflection, might seem a more natural assumption from the physics point of view.)

A variation on this problem that avoids this ambiguity, is to exclude dark points of measure zero, and then ask whether a point source can illuminate the entire interior up to points of measure zero. This is listed as an open problem in the 2008 collection of open problems in computational geometry, by Eric Demaine and our own Joseph O'Rourke.