Is our data "Big Data" (Startup)
Yes, this is a how-long-is-a-piece-of-string question. I think it's good to beware of over-engineering, while also making sure you engineer for where you think you'll be in a year.

First I'd suggest you distinguish between processing and storage. Storm is a (stream) processing framework; NoSQL databases are a storage paradigm. These are not alternatives. The Hadoop ecosystem has HBase for NoSQL; I suspect Azure has some kind of stream processing story.

The bigger difference in your two alternatives is consuming a cloud provider's ecosystem vs Hadoop. The upside to Azure, or AWS, or GCE, is that these services optimize for integrating with each other, with billing, machine management, etc. The downside is being locked in to the cloud provider; you can't run Azure stuff anywhere but Azure. Hadoop takes more work to integrate since it's really a confederation of sometimes loosely-related projects. You're investing in both a distribution, and a place to run that distribution. But, you get a lot less lock-in, and probably more easy access to talent, and a broader choice of tools.

The Azure road is also a "big data" solution in that it has a lot of the scalability properties you want for big data, and the complexity as well. It does not strike me as an easier route. Do you need to invest in distributed/cloud anything at this scale? given your IoT-themed use case, I believe you will need to soon, if not now, so yes. You're not talking about gigabytes, but many terabytes in just the first year.

I'd give a fresh team 6-12 months to fully productionize something based on either of these platforms. That can certainly be staged as a POC, followed by more elaborate engineering.