MS Azure Cloud based services are based on industry standards like SOAP, WCF, WS*, REST etc and as such are technology agnostic and therefore accessible by a PI (Any LINUX System actually)
the trick is do you know how to make it all work, code the client side of the interface etc. I am sure there are already plenty of examples available via google or Bing to help
but at the end of the day, the answer to your question is YES, a PI can be coded to access MS based cloud services.
Kyle, servers on the MS Cloud (Azure)are just like servers any where else and your Raspberry Pi or any other machine will have just as much access to them. The only limitation is what protocols your application supports.
