Webapp: no. Last time I went looking for online pianos was something like five years ago, and for sound and playability, virtualpiano was the best at the time. The search was so disheartening that I haven't gone looking since. A lot of what I encountered was monophonic, or extremely poorly sampled, or click-only, or had awful keyboard support. Like, you'd press a key, and half a second latter the note might press.
It also might press five times at once, making a loud, distorted note that would make me regret wearing earphones. Heh. Actually, virtual piano has a fairly large bug as well: it doesn't seem to have code that only executes when a key is pressed, so if you hold a note down, after about half a second, the script will behave as if you're mashing that key over and over again at something like 0.1 second intervals.
Which is fine, really. I mean, it's free! There's no expectation of perfection. I just wish that someone had put together something more refined...
I forsee a new repo in my future. Damn it. Sorry, let me try to answer the spirit of your question as best I can.
If you or anyone else is interested in the canonical key layout style used to allow qwerty keyboards to play music, I would suggest downloading [LMMS](lmms.io). It's open source, and free. Fire it up, and go to the vertical ribbon of icons on the left. Choose the first. Drag and drop a Triple Oscillator onto either the Song Editor or the Beat/Baseline editor. Once you do, a new entry named Triple Oscillator will be in the editor. Click that, and a window will pop up. Once the window is open, you'll be able to play the keyboard with your qwerty keyboard. If you want, you can make it sound like a piano: grab a copy of the Dead Duck Free Instruments Bundle, go from Triple Oscillator to Vestige and load the DPiano-A.dll extracted from the archive into vestige, in the window that opens when you click its entry in the editor. It's not the best sounding acoustic piano VST out there, but it's definitely the best free one I know about right now.