A couple of months back, Instagram added the ability to view Stories on its mobile site. But regarding the ability to post from the mobile site as well, the blog post simply gave the standard "in the coming months." Now, that functionality is here, along with the ability to save posts.
left: Before. right: After.
To post a Story, simply hit the camera icon on the top left of the site. This was formerly occupied by the 'Discover People' button, but that's now been relocated to the top right. You can take a photo or choose one from your library, then add text or change the text color (it's really advanced stuff). Hit 'Share to your Story,' and you're done. You can also save posts by pressing the little bookmark icon on the bottom right of every photo.
If Instagram adds messaging to the mobile site (which I personally feel is pretty unlikely), then virtually everything on the app will be doable from the site. For now, I'd just like the issue with having to retap on the cursor while tapping every few words to be fixed.