How can I implement the Playsound() function in InTouch Webclient to play audio files?

You are correct!
Playsound() will play a sound from on a InTouch Web application, but it will play it on the server (!)

My bad for not noticing this, that's the downside of testing in a local lab environment.
It will play even if the server does not have any application running. A fun effect, but not so practical.

I have updated my original post, and as noted, the result is that its not possible to do this using the PlaySound() fuction if the intention is to use a web client to access the application.

And as Rickard noted, the mechanism conflicts with the web based server-client way of developing.

Perhaps a workaround is to create a webwidget that supports this, but still, no native solution available at the moment.