Is it DIYable? Yes. No. Maybe?
You may find that you start the job, everything works out just right, and you complete the work in record time and under budget. More likely, you'll run into a bunch of unforeseen problems, take 5 times as long as you'd expect, and end up way over budget.
The best way to tackle DIY work that you are not sure about, is to have the resources available to quickly fix mistakes. In your situation, it might be a good idea to purchase a replacement window before you begin. If you damage the old window beyond repair, you'll simply have to install the new one. On the other hand, if the repair works out, you can return the new window you purchased.
It also never hurts to have somebody with knowledge in the type of work you're doing, so if you run into trouble they can offer guidance.
If you come across something you are really not comfortable with, there are always professionals available to do the work for you. You might also be surprised that they charge less than you might think (not always the case, but does happen).
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.
Best Answer
Pretty sure the sash is stuck within the frame.
However, there might be glazing stops that you can gently pry off with a stiff putty knife. (This is hard, btw. Every one I've ever met has been thin and easy to mangle.)
If you get the stops off successfully, you can order a new thermal unit from a specialty place. (While you're there, show them the stops to see if they have new ones.) Install is pretty straightforward, though you'll be happier if you have a couple of suction cup grabbers.