JSON functionalities are now generally available in Azure Sql Database! All functions that are available in SQL Server 2016 are also available in Azure Sql Database.
Azure Sql Database enables you to get values from JSON documents using JSON_VALUE function, modify values in JSON text using JSON_MODIFY function, transform JSON to table using OPENJSON function or format Sql query results as JSON text using FOR JSON clause:
You can also index JSON values using NONCLUSTERED or FullText Search indexes.
JSON is available in all service tiers (basic, standard, and premium) but only in new SQL Database V12. You can see quick introduction here or more details in Getting Started page. you can also find code samples that JSON functions in Azure Sql Database on official Sql Server/Azure Sql Database GitHub repository.
Note that OPENJSON function requires database compatibility level 130. If all functions work except OPENJSON, you would need to set the latest compatibility level in database.