One alternative is to set up a local web server, as the OP already stated.
The other – which IMHO is faster and less harassing – is letting the extension have access to file URLs, which is disabled by default.
Simply go to chrome://extensions and leave the “Allow access to file URLs” box checked for Vue.js devtools.
Sources:
https://github.com/vuejs/vue-devtools#common-problems-and-how-to-fix