The 'service bureau' does what you're talking about, and has done for a very long time.

https://en.wikipedia.org/wiki/Service_bureau#Histories

I have no idea about the current level of activity in this sector (mainframe proceesing as a service), but would not be at all surprised to find it greatly diminished from it's heyday when dinosaurs roamed the raised floor.

I don't think it takes a huge leap to draw a connection between these businesses and the cloud providers of today.