How can I remove this tile window stool without compromising the installation of the vinyl window?
All windows I've seen were supported at least in part on the sides by nail fins or directly through the frame. Even if your window is technically bottom supported (I can't imagine there is such a vinyl window) there should be enough support from surrounding finish materials to remove the tile and fill in the gap with another material without the window moving in the short term. It may eventually settle if it were intended to be bottom supported and left without support, but it would take a while for the surrounding materials to give up their grip.

You will not have a problem removing material under the window.