Hey Tobias,
You haven't said exactly what it is that you are trying to store. For different data, there are different approaches. I can think of a couple of different cases:
1. I'll start with stating the main idea behind cockpit - it is just UI around other tooling, so ideally that other tooling would deal with it and cockpit would just read/manage it. Such example is for example how we would "store" user preferences for kernel dumps. We read the current settings from /etc/kdump.conf and show that on the page. When a user selects different options, we just tell kdump what the user wants and it updates this config (or we might edit that config directly, not sure now). Cockpit in this case is really just an intermediator.
2. You want to store something about how the user uses the UI itself. This is for example how you would select the language or color of 'Terminal'. This is saved in Cookies (for example look for `theme_cookie` in the source code to see how it is being used). There are other browser storages available that you can use. They all differ for how long the settings are saved. What they have in common though is that they all are only for you current browser and if you connect to the same machine from somewhere else or with a different browser, you will lose these settings.
3. There are cockpit configuration files. These are not really what you want for your extension, so moving swiftly on, but you can read about it in [1]
4. If you still haven't found what you need you are left to designing your own storage. The same way how you would create some config for your command line tool, you can create such config for your cockpit extension. About where on the system the file should be located is very dependent on what you are trying to achieve and SO can always be helpful with that :) [2]
Hope this was to some use.
Happy Hacking,
MM