Azure allows you to store data in any way. As @Gaurav stated, you can use a table repository that allows you to put any data types in your properties (feel free to store JSON strings in your properties) and new repository updates , you can even get the contents as JSON. For more content, you can store objects in the blob store and reference the URIs from your table entities.
If you want something like MongoDB, you can install MongoDB in a virtual machine (where you will incur the costs and maintenance of the virtual machine) or take advantage of a hosting provider such as MongoLab, which offers free and special MongoDB offers through the Azure Store and through their own website.
- Azure, , ( , MySQL ).
EDIT Feb 2015, . DocumentDB Azure (, JSON).