What is the best way to create a database diagram for SQL Azure without having to install SQL Server?
UPDATE: The database diagrams were added back into SSMS 18.1(link)

UPDATE: the database diagram feature 


  Deprecated and removed features in 18.0 (GA)
  ...
  Database Diagrams
  https://docs.microsoft.com/en-us/sql/ssms/release-notes-ssms


The feature was also removed from Visual Studio 2019 SSDT.

Original Answer

The latest version of SSMS (as of v16) supports creating database diagrams for Azure SQL databases. You can download it here:

https://msdn.microsoft.com/en-us/library/mt238290.aspx