As far as I am aware, there are no developments on this front, and I wouldn't hold my breath.
There are two main possibilities for a "native" solution (i.e. one that is baked into Stack Exchange). One is a WYSIWYG, graphical interface which lets you literally draw molecules. I expect that this is way too much work and will never be done just for one site on Stack Exchange. The other is a text-based method where you type in some code and it's rendered as a molecule. For example, the
chemfig package in LaTeX does this. The major difficulty in this would be to make it work with MathJax (note MathJax ≠ LaTeX). However, this is quite difficult to use, and is decidedly not user-friendly at all, so I don't see huge benefits in going down this path.
What I could see being done is a userscript, which adds a button to the Stack Exchange editor which launches an external WYSIWYG popup. The drawing functionality can be provided by some external website (I don't know what's on the market). Then when you hit save, it could generate an image, upload it for you to i.stack.imgur.com, and maybe even put the chemical name / SMILES / InChi etc. as the alt text. From a technical perspective I don't know if that's possible, nor do I have any real interest in looking into it, but I'd view it as the least impossible route. Of course, this is not native, and you'd have to install it on every browser you use.
Personally I'm happy to stick to image generation via ChemDraw (while I still have access to it).