just wondering if the disk space the comes with App Service packages is only for files related to running the web application, or if it can be used for storing user uploads like images and files?
Or do you have to get extra storage if want to store user uploaded content like images, files, etc.?
You can choose to store whatever you want within your Web App's allocated disk space. It's durable storage, meaning it is replicated and won't go away unless you delete the files or completely decommission your Web App (e.g. delete the Web App).
You will need to manage your disk space carefully, since it is capped, based on the App Service tier you choose.
Also: Storage in a Web App is shared across instances of the Web App. If you use a local database such as Sqlite, for example, all instances of your web app would have the same access to that sqlite file.
Note: There are plenty of examples where apps are storing local databases in Web App storage (such as the ghost blog and others).
You can, of course, utilize Azure Storage as well (which scales far beyond Web Apps, and is accessible independent of your Web App, offering additional features you can read up on). The only thing you cannot do with Azure Storage is mount an Azure Files storage area to your Web App (you would need to use an SDK / REST API calls to work with files in an Azure Files space).