That does make sense.
My understanding is that Termianl A collects data and dumps it into a database. Your application reads from that database and generates information for the end user. Since Terminal A and the customer's network would both need internet to be able to see and work with each other, is there any reason that Terminal A can't just have a VPN connection back to the customer site? This way you could host the SQL Instance on Terminal A and the end user could use your software on their own machines and as long as everybody had internet access, you'd be fine.
This would remove the reliance on the cloud and should meet all of your requirement, no? Even if you hosted the SQL instance on the cloud, you would still need to open ports in the firewall, but it would be all internal to their network due to the VPN. And there are free VPN solutions as well as paid ones that all offer different things.
One issue I do see with the cloud is, lets say you pick the cloud company called "Clouds" (an example one, I don't know if they actually exist or not). The solutions works good for a few years and you gain a few more customers. Suddenly the company closes without notifying anybody. All of your customers are left hanging.
This type of scenario happened to me with 2 different cloud file hosting services I had signed up to. I am thinking about companies like megaupload. Their services ran great for a number of years and then one day they all got shut down. They came back up with a different name, but a lot of their old content was lost. Mind you they wouldn't be what you would be looking for, but just as an example. The 2 other companies I was referring to were not megaupload mind you...
Also, in your scenario, what happens if the internet cuts out? Does Terminal A stop collecting data while waiting for the internet to kick back in or does it cache it locally or does the data get lost?