The minimum permission required to run a user creation script for Azure SQL is Server Admin rights.
For more information about the characteristics of the Server admin and Azure Active Directory admin accounts, review Controlling and granting database access to SQL Database and SQL Data Warehouse.
The Azure SQL recommended permissions for managed instances in SQLDM are the same as On-Premises recommended permissions.
Minimum Permissions Azure SQL
SQL Diagnostic Manager now includes a SQL script to create a new user with minimum permissions for the product to function. This script has two sections, which must be executed accordingly.
Replace all instances of '$username' with the name of the SQL account that you want SQLDM to use to collect diagnostic data.
Replace '$password' with the desired password for the SQL account.
Execute the first section of the script against the target monitored Azure SQL Database using an account with server admin rights.
Execute the second section of the script against the target database using an account with server admin rights.
Find the RecommendedPermissionsAzureSQLDB.sql script below:
Review the Minimum and Recommended Permissions for Azure SQL for more details on the permissions you need to create a new user for Azure SQL Database.