One of the most common queries that Instagram users make is the possibility to put links or links to other websites in their stories.
To link a Web site from a storie must have a verified account of Instagram, users with verified accounts have at the top of the screen a hyperlink icon and there you can include the link.
When you add a link to your story, those who access it will see the option view more at the bottom of the screen and can touch it to see the link inside the application.
If you don't have a verified account you won't have the possibility to add links in your stories, and the only sites where you can add links will be in your Bio and through direct messages.
Verifying an Instagram account is not an easy task in fact there are many users who have verified accounts on Facebook but fail to verify their account in Instagram.
To verify your account in Instagram you must prove that someone is impersonating your identity or usurping the name of your organization or company and complete the complaint form you find in this link https://help.instagram.com/contact/636276399721841
Update May 2018
We update this post with new information referring to this topic. Instagram is currently testing the option to share links from our stories not only with verified account users but also with any user that exceeds 10k followers.
So if you have more than 10000 followers in your Instagram account you should be able to add links to your story as shown in the image here:
Instagram is expected to finally put this functionality at the disposition of all users. But as in many other cases this is done in a staggered way