Are you referring to an actual MathJax extension or simply the esint LaTeX package? We're not aware of such an extension for MathJax so it would be good to know what you are referring to. MathJax is not (La)TeX and thus packages for (La)TeX have to be re-implemented as MathJax extensions.
Looking at esint, I think most of these (all except \landupint and \landdownint?) could be handled by Unicode and thus MathJax's non-standard
\unicode macro. However, for most of them our default MathJax (CM derived) fonts do not provide glyphs, so you'll need to switch to another font, see
http://docs.mathjax.org/en/latest/font-support.html for the available options. I'm not sure which ones provide enough glyphs though so you'll have to test that.
Peter.