[TheBang]

This is what I've been using, and it works well on vB3 and vB4. It supports video, as well as the caption, and is not a fixed size like the iframe method at the beginning of this thread.

Title: Instagram
BB Code Tag Name: instagram

Replacement:

<blockquote class="instagram-media" data-instgrm-captioned data-instgrm-permalink="https://www.instagram.com/p/{param}/" data-instgrm-version="8" style=" background:#FFF; border:0; border-radius:3px; box-shadow:0 0 1px 0 rgba(0,0,0,0.5),0 1px 10px 0 rgba(0,0,0,0.15); margin: 1px; max-width:658px; padding:0; width:99.375%; width:-webkit-calc(100% - 2px); width:calc(100% - 2px);"><div style="padding:8px;"> <div style=" background:#F8F8F8; line-height:0; margin-top:40px; padding:33.33333333333333% 0; text-align:center; width:100%;"> <div style=" background:url(data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACwAAAAsCAMAAAApWqozAAAABGdBTUEAALGPC/xhBQAAAAFzUkdCAK7OHOkAAAAMUExURczMzPf399fX1+bm5mzY9AMAAADiSURBVDjLvZXbEsMgCES5/P8/t9FuRVCRmU73JWlzosgSIIZURCjo/ad+EQJJB4Hv8BFt+IDpQoCx1wjOSBFhh2XssxEIYn3ulI/6MNReE07UIWJEv8UEOWDS88LY97kqyTliJKKtuYBbruAyVh5wOHiXmpi5we58Ek028czwyuQdLKPG1Bkb4NnM+VeAnfHqn1k4+GPT6uGQcvu2h2OVuIf/gWUFyy8OWEpdyZSa3aVCqpVoVvzZZ2VTnn2wU8qzVjDDetO90GSy9mVLqtgYSy231MxrY6I2gGqjrTY0L8fxCxfCBbhWrsYYAAAAAElFTkSuQmCC); display:block; height:44px; margin:0 auto -44px; position:relative; top:-22px; width:44px;"></div></div><p style=" color:#c9c8cd; font-family:Arial,sans-serif; font-size:14px; line-height:17px; margin-bottom:0; margin-top:8px; overflow:hidden; padding:8px 0 7px; text-align:center; text-overflow:ellipsis; white-space:nowrap;"><a href="https://www.instagram.com/p/{param}/" style=" color:#c9c8cd; font-family:Arial,sans-serif; font-size:14px; font-style:normal; font-weight:normal; line-height:17px; text-decoration:none;" target="_blank">A post shared by</a> on <time style=" font-family:Arial,sans-serif; font-size:14px; line-height:17px;" datetime="2017-01-01T19:00:00+00:00">Jan 01, 2017 at 12:00pm PDT</time></p></div></blockquote>
<script async defer src="//www.instagram.com/embed.js"></script>

Example: [instagram]BoW_r_ZBqhD[/instagram]

Use {option}: No
Remove Tag if Empty: Yes (and everything below this option set to Yes also).