It's certainly possible, but it'll require additional hardware. Once the RPi loses power, that's it. Just like your desktop, there's no magical way for the RPi to do anything (like send a notification) once the power is gone.
You could add a small UPS to the RPi. There are a few that are even designed specifically for the RPi, like this one. If the power goes out, the UPS will send a notification to the RPi. Then the RPi can handle sending a message across the network. If you're fast enough, the RPi may never experience downtime. If you're not fast enough, the RPi can gracefully shutdown.
Alternatively, you could add microcontroller to monitor the power and send a message if needed. Something like this should be able to run off a battery for along time.
I think the best and simplest option though would be to ping the RPi on a set schedule. Since your RPi is a network attached kiosk, you obviously can contact it through said network. Why not just setup a handshake every 1/5/60/whatever minutes? This could be as simple as a script on your server or even desktop workstation that just pings a list of IPs/Hosts and then notifies you which ones don't answer.