If you have access to the build output of the Vue.js UI it can be quite straight forward in ASP.NET MCV. I don't now how applicable to webforms it will be but here was my solution to the issue:
@{
Layout = "~/Views/Shared/Layouts/_Public_Injectable.cshtml";
ViewBag.Title = "Page Title";
}
@section scripts{
@Styles.Render("~/content/css/vue")
}
<noscript>
<strong>We're sorry but this page doesn't work without JavaScript enabled. Please enable it to continue.</strong>
</noscript>
<div id="app"></div>
<script src="~/Scripts/Vue-Build-Files/js/chunk-vendors.738367a5.js"></script>
<script src="~/Scripts/Vue-Build-Files/js/app.40933a0b.js"></script>
Essentially the solution is to take the Vue.JS build output and add it as a set of scripts to the page and ensure that your page has the necessary index.html to work with the vue application