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).

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).