It seems that web-mode has supported Vue.js since May 2017, the "Venora Release".
Configuring Emacs to handle Vue.js files (extension .vue) is as easy as writing
(add-to-list 'auto-mode-alist '("\\.vue\\'" . web-mode))
in one's .emacs file.
But web-mode does not like the <template> tag. What is the right .emacs setup/config?