dramamama's question actually might be a great starting point for you. Why are you writing to her? Do you have a particular connection to her story, her hometown, anything like that? Does OOTI or the role of Ti Moune mean a lot to you, and why? Do you just want to wish her luck and best wishes? Any of those things are lovely thoughts and things that most actors I know would be delighted to read in a letter. Introduce yourself, and then explain why you're writing to her/what you want to tell her.
Try to remember that she's a perfectly ordinary person, just like any of us, who just happens to have a profession that puts her in the public view more than, say, an accountant. Just write what you feel, don't worry about the "right" thing to say. I know anxiety can make things like this feel really big and intimidating, but please know that most actors, especially ones early in their career, are thrilled to get fan mail and genuinely touched to read what their fans say.
If you want a reply, include a self-addressed stamped envelope- tuck it and your letter inside of a larger envelope, and address that outer envelope to her, care of the stage door of Circle in the Square. Don't stress if it takes a while to get a reply - some actors/theatres are really fast with mail, others are slower.
Finally, I wouldn't suggest asking questions like this on these boards. You'll get a lot of snark and condescension. Just remember that a lot of the very people who are on Broadway and behind the scenes were once nervous theatre fans too - you're in perfectly good company, and being a fan is nothing to be ashamed of! (There are some fans, especially of certain shows/actors, who take things way too far, but just exercise basic manners and common sense, and you'll be fine). Good luck!