Whenever I try to buy something my cursor changes from the styled one in Dota 2 to the normal one from Windows and if I move it around I see it change to select text, so CTRL + A gives this output. [pastebin link]
Basically the Steam popup/window is invisible. I tried reinstalling Dota 2 but this didn't fix it.
Best Answer
Try relaunching Dota 2 from the Big Picture Client. The result is that a different Steam layout appears whenever you Shift-Tab. See if you are now able to purchase.