Full disclosure - I work for Eagle Eye Networks.
The Eagle Eye system, which is cloud based, is able to do this. In fact we have several customers using our system for exactly that. We also have a couple of customers using it for dog day care too.
Because users on our system connect to the cloud to view video, not the individual site, the local Internet connection only streams the video once, regardless of how many people view it.
We also have fairly granular permissions, which allow you to grant access to end users on a camera by camera basis. For example, if you have kids in the 3 yr old classroom, you can only see those cameras. But someone with kids in the 3 yr old and 4 yr old classrooms can see both classrooms.
We have free iOS and Android apps that can be downloaded from the respective stores, and there is no per user charge. The day care would just pay based on the number of cameras, and how long they want to store the video.
We didn't design our system specifically for this use case - but it does fit pretty well.