I have been using VSCode for teaching the last few years, and routinely use a VSCode without plugins for that purpose, and Emacs keybindings work fine in its text area (the code editor itself). If it doesn’t work for you, then apparently something is broken on your computer. Not all keybindings work but the ones listed under Text Editing here certainly works for everyone else: https://support.apple.com/en-us/102650

I am typing this in a textarea in Chrome, and the basic Emacs navigation commands all work: C-a, C-b, C-e, C-f, C-n and C-p. (The M- equivalents don't, though. You just get some special chars: å, ∫, ´, ƒ, ñ (dead char for the accent), π.)

Chrome Version 149.0.7827.155 (Official Build) (arm64)

Very similar in Visual Studio Code.

(I started using macOS in 2010, I think, and the Emacs-style shortcuts have felt pretty well supported the whole time. I can't promise that applies to Chrome as well though, as I'm not a regular user.)

Yes. They work fine in Chrome. Mellel is the only app I can think of where they don’t work.