I know Vue, but do you really need Node.js on the server to use Vue?
As far as I remember, Vue is just a frontend framework (like a suped up version of jQuery). Which means that any hosting that can share static Javascript files and HTML can be used to host it. Which is no problem here.
You will probably need Node.js and NPM to build the Vue code to something that can be used in browsers. But you can do the building on your own computer and upload the compiled assets to your hosting account.
So, to answer your original question: where do you want to install Node.js? If you want to install it on your own computer then yes, probably. If you want to run it on our hosting service, then no.