There must be a combination of Metra (train) and Pace (bus) that will work. It is walkable from the Oakbrook Terrace shopping mall, which is huge, so you could go to their websites and plot it out, or call the theater and maybe they would have some advice. I've only ever driven there, sorry!
This isn't relevant to your question, but I stumbled across this production last Friday evening and really enjoyed it! I was staying the night in Chicago to see Caroline or Change at the Court Theatre (which was EXCELLENT), but I had no idea the Drury Lane Theatre was practically connected to my hotel. Barbara Robertson's Mame was very good, but I thought the best performance was Alene Robertson's Vera Charles.