I first hit a case where the Esc key did nothing while editing with Vim in the web console of a guest OS on ESXi.
I could not leave insert mode, and of course the other keybindings did not work either.
I looked around the web in case others had seen the same thing.
As in that article, Shift+Esc and Ctrl+[ also failed to return me to Vim command mode. As a last resort I pressed Ctrl+c, got Type :quit<Enter> to exit Vim, and was able to save with :wq.
The same symptoms appeared no matter how many times I tried.
I found that Vim commands work if I click the mouse immediately inside the ESXi virtual machine console, so this is probably happening only under that specific condition.