You could use SQL Server 2012 Data Tier Application Framework API to do that. You could download it from
http://www.microsoft.com/en-us/download/details.aspx?id=34964. In that framework, you could use DacServices::ExportBacPac to export SQL Azure database and import using DacServices::ImportBacPac.
Besides, there is no API which could export the SQL Azure database directly to Azure BLOB storage, there need to be a local temporary output. You could refer to the following sample,
DacServices dacService = new DacServices("server=<username>;uid=<user name>;password=<password>");
dacService.ExportBacpac(<file path>, <database name>);
BacPackage bacpacFile = BacPackage.Load(<file path>, DacSchemaModelStorageType.File);
dacService.ImportBacpac(bacpacFile, <database name>);