If you plan to use Tryton as back-office, of course it makes sense to use it also as backend for your web application in Vue.js.
I also made many Vue.js application using Tryton as backend using User Application API or even just directly route and web_user
module.
You must know that there is a plan for REST API for user application.
About Vue.js, I personally tries to avoid to deal with node.js and dependencies. Instead I follow Use Vue to create a SPA without any Node modules - DEV Community and use importmap. It generate a little bit more requests but not that much and mostly without extra delay thanks to http2.