Is it okay to send my MA thesis for an internship I'm seeking to apply for (they asked for it among other documents) even if the thesis is not published yet?
If the status is 'finished but not published/defended/graded/...' then send it with the notion that is hasn't been published yet but is finished as a whole.
If you're still writing the thesis or waiting for feedback of your supervisor, then send it with with a note that mentions this fact and the expected finalization date, and possibly check with your supervisor if they expect major revisions will be needed.
If you are the sole author and you still hold copyright (likely since it isn't published), then you can send it. It is your work and you own all rights.
However, whether it is wise to send it or not is another question. I suggest that you talk this over with your advisor so see what the downstream consequences might be, especially if you intent to extend this work in the future. Some work should be held confidential for various reasons and you should get some advice on that.
It may be that a brief abstract or description of your work might do just as well for the internship and you might also explore that possibility.
But get some advice from someone more experienced who also knows your work.
If you are not the sole author, then you need permission from the other authors.