I think reservations open 11 months out, but I'm not sure that's true.
We're going this July (first time, can't wait!). I wrote it on my calendar so I wouldn't forget, but I poked around and didn't call for about two weeks after the dates opened. I got a cabin, but it wasn't the style I wanted! I wanted the newer Western cabin, but I had to settle for the older Pioneer cabin (though I did snag the very last one with a rim view). And this was at the North Rim, the less popular side -- I'm sure it's worse at the South Rim.
Along those same lines, if you want to take a mule ride into the canyon, you have to make those reservations well in advance.