Is there any API for Sublime text's package manager? I tried to find it but couldn't find any mention of it anywhere.
Yes, every data-driven page on https://packagecontrol.io can have .json appended to the end to get the raw data. Here are some examples:
This is how the website is implemented – there is a JSON API and a JS SPA that queries the JSON API and uses Handlebars template to create the page. On initial page load, the JSON is rendered using Handlebars on the server so there is no initial "loading" screen.
For pages that are not data-drive, i.e. documentation pages, you can append .html to get the primary content:
Additionally, the whole website it open source at https://github.com/wbond/packagecontrol.io.