According to this rather old, but still current page on Apple's Developer Site (the public portion), there are supposed to be APIs within iOS 6 that allow for access to "real-time video stabilization". Sadly, I cannot find any API documents that seem to support this.
Could someone please point me in the right direction?
Well, I finally found it.
Discussion
The default value of this property is NO. If the supportsVideoStabilization property contains the value YES, you can change the value of this property to YES to enable video stabilization for the connection.
You can only enable video stabilization for connections that operate on video. Even so, not all source formats and video resolutions are supported. Enabling video stabilization may also introduce additional latency into the video capture pipeline. To detect when video stabilization is in use, use the videoStabilizationEnabled property.