You want to replace the leaky, drafty original basement windows with glass block windows.
You can do this.
I assume this fabricator will assemble the window unit, and then you intend to install it as a single unit in the opening. The point I’m making is that you won’t be laying it block-by-block in the window opening, as if laying brick.
First, please check and see why your present basement window is leaking. I know that it’s a bad window and is probably cracked and such. But water should not even get close to a basement window. So, even with a cracked window, you should not be getting water into your basement.
Often it’s as simple as making sure that a downspout isn’t pouring water around the window area. Or you might just need to re-grade the ground around the window, sloping it away from the house.
So, yes, you can install a glass block window yourself. The fact that it’s glass block is inconsequential; installation would be just the same as any other type of window.