No, it isn't possible to do it through the cloud in the way you have described. It is possible to offload transcoding to the cloud (changing from one file format or size to another) however you can't use that cloud for capture hardware as that needs to be physically connected to the source. There are no services that I know of that will run a satellite box for you and hook it up to capture for you, so the source would have to be at your satellite box.
Also, while it is possible to have transcoding done online, to stream 25 uncompressed satellite channels over the Internet to a cloud service provider for transcoding would require a MASSIVE amount of bandwidth to be available. You really don't have any option other than local transcoding for that kind of thing in order to get the bandwidth requirement down far enough to have streaming 25 channels to become viable.
Even then, you would have a hard time streaming 25 channels in HD quality without a massive upstream Internet connection. Even 25 SD streams would be require more than most common ISP connections would support.