SQLite on Azure website
Based on my experience if you want to use SQLite with Azure Websites you can keep the database file within your deployment package so it will stay at the same server where your website is. Azure websites provide 1GB of application storage which is plenty for a database file. Your content with the websites will persist and access to SQLite DB will be fast. This is super easy and you can very easily do with ASP.NET web application or any other.
The problem of choosing Azure Blob storage is that if the database file is stored at Azure Blob storage, there are no API that SQLite can write to that file. So one option you could have is to writing locally first and then syncing to Azure Blob storage back and forth while others on SO may have some other options. If you want to backup your database file to Azure Blob storage for any reason you sure can do that separately however I think if you choose the have SQLite, the best would be the keep the database file with website to make it simple.