I have paid for software through the software center that says a mouse is required but no mouse is setup, I am running 12.04 on my laptop.
It requires a pointer device. You can use a mouse, trackball, trackpad, or similar device to interact with the app. It doesn't absolutely have to be a mouse.
If however, you only have a touchscreen, and no other pointer device, you may find it difficult to use some features of the software.