Signalfade

Adds a new type of feed you can subscribe to. 
 http://example.com/feed/json or <code>http://example.com/?feed=json</code> to anywhere and you’ll get JSON back.

JSON Feeder

Adds a feed based on the jsonfeed.org standard that one can subscribe to or parse.

David Shanske

Daniel Jalkut (Red Sweater)

mantonr

Adds a JSON Feed to your WordPress site by adding <code>/feed/json</code> to any URL.
The JSON Feed format is a pragmatic syndication format, like RSS and Atom, but with one big difference: it’s JSON instead of XML. Learn more at <a href="http://jsonfeed.org/" rel="nofollow ugc">jsonfeed.org</a>.

JSON Feed (jsonfeed.org)

This plugin is simple to install, adds no weight to your site, and does what it says on the tin.

JSON Feeder vs JSON Feed (jsonfeed.org)