Most of the Azure services have a REST API for them. My experience was most of the hard work is wrapping your head around the Azure REST API and the requirements for things like authentication (Shared Key/Shared Key Lite) and specific headers (continuation/pagination).
For example, here is the Azure Table Storage REST API:
https://msdn.microsoft.com/en-us/library/azure/dd179423.aspx
Once you get that figured out, the SmartApp methods httpGet, httpPost, httpPut, etc are fairly straightforward to use.
PS. I posted an article on using Shared Access Signatures with Azure Queues which may be of interest to you if you go down the Azure services path:
http://boshdirect.com/tech/azure-queue-via-shared-access-signature-and-rest
PPS. I would also recommend testing all of your REST API calls in something like Postman. Once you get your API calls figured out, then look at moving things into SmartThings. 