If you are accessing this course on the desktop or a laptop, Google Chrome (currently the most popular browser in the world) is the recommended browser for this site, and switching to it will likely solve any issues you may be experiencing. It is a free download, easy to install, and available for all platforms.
Some third-party extensions for web browsers, such as ad blockers, might interefere with the typing functionality. If you are using such an extension, turn it off temporarily and see if that is the cause. If so, it should be possible to whitelist this site so that the extension is turned off for this site only.
This course is not designed to be used with a soft keyboard on a tablet, although it has been successfully tested to work with iOS devices (iPad and iPhone). If you are on Android or a Windows Mobile device it is hit or miss, however it should work absolutely fine with an external keyboard. It is highly recommended to learn touch typing on a physical keyboard for the tactile feedback, and bluetooth keyboards can be obtained very inexpensively these days (i.e. as little as $10). This is a VERY worthwhile investment to learn touch typing, a skill which will last you a lifetime.
If you're still having problems, you can still access the old, Flash-based version of the course here.