Will the theme work without purchasing the licensing for Sublime Text 2?

Sublime Text is a text editor. It has nothing related to WordPress Themes. It isn't required to buy Sublime Text :)

But it does require (it isn't required, but your supposed to because it is very important for later in your course) any text editor like TextEdit (if you are on a Mac this is already on your computer ) or Notepad++ (which is already included on Windows).

Also you can search "text editors" on a web browser and you can find some great, free/paid text editors :)

P.S.: If you really like Sublime Text, actually you still can use it because you actually can keep using it because the trail is infinite :)

Good luck and I hope this helps! ~Alex