It looks like you no longer need to have Instagram app installed on your phone to access the 24-hour disappearing Stories feature. Instagram recently announced that they are bringing two new experiences to Instagram.com mobile web version: creating stories and saving posts you like.
When you visit Instagram.com on your mobile web, you will see a new camera icon on the top left of your feed. Just tap the button to take a photo or to add a photo from your photo library, and you can share it as an Instagram Story; you can also do some basic editing like adding text, and changing the text colour. Once you’re done, just post it to your story by tapping on “Share to Your Story”. Sadly though, features like face filters and stickers are not available; you also can’t post videos to your Instagram Stories via Instagram.com.
In addition to that, Instagram is also adding the ability to save posts you like so you can visit it again later. Tap on the bookmark icon on each post to save them. You can access saved posts by tapping on the bookmark icon in the top right corner of the page.
Instagram says that these new features will be available to everyone in the coming weeks, but it is already available for us.