On Aug 7th, 2019, WordPress closed JSON API Plugin from wordpress.org repository and no longer available for download and which means you can’t install it from WordPress Plugin installation page,
JSON API WordPress Plugin has been a great way to use WordPress as backend for your JS or Mobile apps, and it is unfortunate that it was not being maintained by its original author. It was not being supported in WordPress forums and ultimately WordPress website closed it down, the exact reason of closure is not yet known but since it has not been updated for last 4 years, this must have been the reason.
All is not lost though, I have copied the original plugin, with some changes to make it compatible with the latest WordPress version, and hosted it on GitHub. so you can download JSON API Plugin from https://github.com/PI-Media/json-api until it is republished and available on WordPress.
I have couple of plugins JSON API User, its pro version JSON API User Plus and JSON API Auth that extend JSON API Plugin and add new functionality and are widely used. I will now add some features and try to republish it on WordPress.org so that it is available directly from within WordPress.
Meanwhile you can download it from GitHub and install it manually. Make sure when you download from GitHub, unzip and rename the folder from json-api-master to json-api and then upload as zipped file named as “json-api.zip” to WordPress using its add plugin, upload plugin option.
You can also download the JSON API Plugin from Dropbox so that you don’t have to rename the folder, just download the zip file and upload to WordPress.
Hope that helps.