Depends what sort of data you want to store. If you want to store things like the broadcasters settings for your extension then your server can still use the Configuration Service https://dev.twitch.tv/docs/extensions/building#using-the-configuration-service
If you want to store data about individual users though the config service wouldn’t have the capacity for that and you would need your own database or storage solution. And keep in mind that user data you store may be subject to GDPR and other privacy laws so make sure you’re legally compliant with how you choose to handle that data.