Scaling Azure SQL database will make my website goes down?
How does the scaling process work? Is there a flow, diagram that
illustrates it?
See this link:
https://docs.microsoft.com/en-us/azure/sql-database/sql-database-service-tiers
Will the scaling cause any downtime in my website?


No new connections for a brief period and existing connections might rollback.


  Changing the service tier and/or performance level of a database
  creates a replica of the original database at the new performance
  level, and then switches connections over to the replica. No data is
  lost during this process but during the brief moment when we switch
  over to the replica, connections to the database are disabled, so some
  transactions in flight may be rolled back. This window varies, but is
  on average under 4 seconds, and in more than 99% of cases is less than
  30 seconds. If there are large numbers of transactions in flight at
  the moment connections are disabled, this window may be longer.



Am I able to write things to the database during the scaling
process?


Yes but you might see your transaction is rolled back.