JSON is available in Azure SQL Database
JSON functionalities that are added in SQL Server 2016 are also available in Azure SQL Database see Public preview: JSON in Azure SQL Database. All functionalities that you can use in SQL Server 2016 RC (i.e. FOR JSON, OPENJSON, built-in functions) are also available in Azure SQL Database, including JSON_MODIFY function that is added in RC version.
JSON is available in all service tiers (basic, standard, and premium) but only in new SQL Database V12. We don't have plans to support it in older versions, since we expect that everyone will migrate to new V12. If your JSON functions do not work, please check version using SELECT @@version.
Also, OPENJSON function requires database compatibility level 130. If all functions work except OPENJSON, you would need to set the latest compatibility level in database. Note that level 120 is set by default even in new databases.