How can I store JSON data on Azure Websites?
EDIT Nov 2019 CosmosDB Mongo API is now available as a native Azure document-database service (e.g. JSON).
Original answer below
Azure will let you store data any way you want. As @Gaurav stated, you can take advantage of Table storage, which lets you put any type of data within its properties (feel free to store JSON strings in its properties), and with the new storage updates, you can even retrieve contents as JSON. For larger content, you can store objects in blob storage and reference the URI from your table entities.
If you want something like MongoDB, you can either install MongoDB in a Virtual Machine (where you'll incur the costs and maintenance of the VM) or take advantage of a hosting provider such as MongoLab, which offers free, shared, and dedicated MongoDB offerings via both the Azure Store and via their own website.
You won't be able to install a storage service on Azure Web Sites, but you can easily access any type of storage described above (or numerous other options, like installing MySQL or other databases).