Is it possible to download the draw.io app to my computer via their web UI?
I need it for my exam and have no internet access - and I haven't found any drawing program that fits my needs like this superb program
Yes, it is called "Draw.io Desktop". I have not tried it myself. I just found it as the second hit when I Googled "draw.io download".
The web app can be used offline with the following link: https://www.draw.io/index.html?offline=1
There is also a Chrome App.