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I looked left and right but it doesn't say anywhere what software is it using to run a smalltalk environment in the browser.

I played with (Pharo) Smalltalk a bit in the past, it'd be nice to try it again in the browser.

Perhaps Lively Kernel?

This is using my Smalltalk-78 Virtual Machine https://github.com/codefrau/Smalltalk78

For a more modern Smalltalk in the browser you can try SqueakJS https://squeak.js.org/