The IndieWeb stack supports JSON in that the primary building blocks coming out of the community are built using Microformats 2. One of the primary features of MF2 is its canonical json representation. An easy way to understand this is that a Microformat 2 parser turns any webpage into a JSON representation of that page. This representation can be optionally enhanced with the use of prefixed html class names added to the html document by the publisher.
Example: Parsed SocialWG IRC logs: http://glennjones.net/tools/microformats/?url=http%3A%2F%2Fsocialwg.indiewebcamp.com%2Firc%2Fsocial%2F2014-10-21&filters=&callback=
Indieauth is an authentication mechanism that uses a personal domain as an identity (instead of email). It attempts to fix usability and implementation issues that OpenID suffered from (and optionally can act as an openID provider). It uses MF2 to parse the provided identity domain for links in the html marked up with rel=me. http://indiewebcamp.com/IndieAuth
Webmention is a pingback but instead uses querystrings to deliver the parameters between sites instead of xml. The verification and vouch process parse web pages using MF2 parsers to get a JSON representation of the document and then uses that information. http://indiewebcamp.com/Webmention http://indiewebcamp.com/vouch
Micropub is similar to Atompub but instead uses microformat 2 vocabulary and delivers it as form encoded data (instead of XML) to allow for html form micropub clients. It also uses oauth2 bearer tokens and indieauth. The micropub endpoint parses these requests and turns them into the canonical JSON representation of the MF2 data within the endpoint. There is ongoing discussion about allowing the optional use of submitting the MF2 JSON objects directly or sticking with form encoding only. It should be noted that there has been more implementations of interoperable MP Clients and Endpoints within its first year than Atompub ever acheived as far as I am aware. http://indiewebcamp.com/Micropub
Hope that clears things up.