I have uploaded a number of inquisit scripts and I can run them online by going to "My Account - Web Scripts."

I would now like to download them onto another computer. However, I can't see anywhere on the page where I could do this. All I am able to do is run the scripts online. 

If this is not clear I am happy to specify. Thanks in advance!

There currently isn't any easy way to do this. Go to the respective script's launch page and inspect its source code. There'll you'll find the script link which you can then paste into your browser's address bar to download the file.