You can take this with a grain of salt, but I've heard that some stutterers do well listening to music when they need to talk. I think it's easier if you can't hear yourself? If that helps you at all, could you wear some headphones with music while you talk? Earbuds wouldn't be too obvious. You could also maybe prepare a "teleprompter," have your speech written on a posterboard that is right beside the camera but out of view, so that you are just reading aloud instead of speaking spontaneously.
I also like the idea of doing it interview style, like a conversation between you and MIL or you and your DH, or doing it in prerecorded segments that will be put together.
Mostly I would just say to prepare ahead of time... know what you are going to say, practice it, get very comfortable with it, and then... let it go. Realize that if it doesn't turn out wonderfully, at least you made an effort to do something really special for your MIL.
Have you tried talking to your MIL specifically about the stuttering problem and telling her honestly that the situation makes you nervous, you don't really want a video preserving your stuttering for all time, and asking if she can work with you to help you participate but not embarrass you? If she's a good MIL, and from your description it sounds like she is, she might be helpful.