I had this very same question when I applied this cycle. Finding out, though, was very difficult.
I didn't find the date (for when AMCAS started accepting the application) on their website. I tried calling and emailing them and got nothing.
They finally posted it on their website in mid-May I believe. It was originally set for June 1st, but there always seem to be delays and, of course, the date got pushed back. I think the acutal date this year was June 10th or something like that.
Basically, as long as you submit it in June, you're ahead of the game and in good shape. They always shoot for June 1st, but they're never actually ready to accept them by then, so usually it ends up being some time in early to mid June.