Instagram has provided its users with a badge whose code can be added to a text widget.
After you log in to the Instagram website, click on your username and then "Badges" from the drop down menu there. Select your badge and copy only the last part of the code. Paste that into a text widget.
<a href="http://instagram.com/<strong>yourusername</strong>?ref=badge" class="ig-b- ig-b-v-24"><img src="//badges.instagram.com/static/images/ig-badge-view-24.png" alt="Instagram" /></a>
Alternately, you can save the badge you wish to use, upload it to your Media Library and then use it in an Image Widget, linking to your user profile on instagram.
Also, there is a discussion here about using the flickr widget for instagram images, but be warned it does not always work as expected. http://en.forums.wordpress.com/topic/issue-with-flickr-widget?replies=23#post-1155866