One option for an 11.1V system is a 'CIM' motor. They're rated at 5300rpm at 12V and not too expensive. The downside is that it would need gearing down, perhaps around 10:1, but that would be true of most scooter motors and axe systems so depending on what tools/experience/workshop space you have, it may be within your means.

In terms of mounting, I believe it has threaded holes in the front face, so you could either drill corresponding holes in a vertical chassis panel or make a 90 degree L bracket to mount it in your robot depending on your design.
A 10:1 ratio may be quite difficult to achieve in a single stage transmission using chains and sprockets, so it might be easiest to use spur gears. I know Lucy (Ocracoke) has done similar with limited tooling so she may be able to provide more detail.

Here's a link to the motor: https://www.robotshop.com/uk/12v-531...RoCUjcQAvD_BwE