Instagram may not offer native RSS feeds, but it is still possible to generate feeds through 3rd party web applications.
Here are three options currently working as of November 22, 2016:
Queryfeed creates an RSS feed for Instagram as well as Twitter, Google+ and Facebook. Visit Queryfeed, scroll down to the Instagram section, enter a username, hashtag or geo-location and hit “Search” to generate your feed. The free version caches results for 3 hours and limits RSS items to a max of 10 per feed.
http://instatom.freelancis.net/username creates an atom feed from your Instagram account. Just replace username with the appropriate username.
RSS-Bridge is an open-source “PHP project capable of generating ATOM feeds for websites which don’t have one.” You can set it up on your own server, or just google “rss bridge” and you’ll find several hosted installations that you can use freely.
For example, rssbridge.buddylist.co is one of the top results. Go there and scroll down to the box labeled Instagram Bridge, enter the desired username, and choose the type of feed you want. Feed options are Atom, Html, Json, Media RSS, and Plaintext. RSS-Bridge also features feed generators for a wide variety of other sites including Facebook, Twitter, Google+, Pinterest, YouTube and more.
Previously listed services that no longer generate Instagram feeds: Websta, Iconosquare and Ink361.