Hi David. I also teach vector calculus, and I looked into this issue some time ago. My conclusions at the time: it looks like MathJax doesn't support esint, and it looks like there's no braille symbol for \oiint. So I decided just to live without it, and use the regular \iint for all of my surface integrals, including those over closed surfaces.
Now, the stuff you mentioned about unicode-math: that's all news to me. So I've subscribed to this thread, and I'll be interested in seeing what the group comes up with. But even if we could use \oiint in the HTML conversion, I'd still be uneasy because of the braille issue. I might never have a student who needs things in braille, but I want to play it safe.
-Joseph