Yes, it is possible to download the app without the Store. Some available distributions are listed in the Manual Installation section of the MS guide (https://learn.microsoft.com/en-us/windows/wsl/install-manual)
The actual manual installation instructions are hidden in the Server Install section of the manual (https://learn.microsoft.com/en-us/windows/wsl/install-on-server), but my understanding is that you can just run the .appx file you downloaded and it should work anyway.
Whether it’s wise to install WSL/Ubuntu on a work computer that you don’t control is another question, but one which only you can answer.