How to use Sublime Text with the built-in Apple spellchecker

Solution 1:

Sublime Text is tried to be cross-platform and as there is no built-in auto-correction functionality in other platforms, so if I was the programmer, I wouldn't relay on a feature present only in one of my target platforms and would implement the feature built-in my program to be portable to other platforms.
I love that feature of OS X too, but unfortunately I don't think if that would be possible in your case. :)

Solution 2:

Old post, but by now a plugin exists that interfaces with OS X's built-in spell checker. See https://www.sublimetext.com/forum/viewtopic.php?f=5&t=11692. Haven't tried it myself yet, but will now!