Reports

Thanks to @moritz-ringler for bringing up stackoverflow.com/a/76934503/4883195. I've managed to put together a working github.com/OnlyLoveOleg/vue3-vuetify-webcomponent example in case anyone is looking for a solution to this problem.

index.html

<!DOCTYPE html>
<html lang="">
  <head>
    <meta charset="UTF-8">
    <link rel="icon" href="/favicon.ico">
    <meta name="viewport" content="width=device-width, initial-scale=1.0">
    <title>Vue3 Vuetify Web Component</title>
  </head>
  <body>
    <my-navbar></my-navbar>
    <script type="module" src="/src/main.js"></script>
  </body>
</html>

src/main.js

import { defineCustomElement } from './defineCustomElementWithStyles'
import '@mdi/font/css/materialdesignicons.css'
import { createVuetify } from 'vuetify';
import { aliases, mdi } from 'vuetify/iconsets/mdi'

// Import the Vue component.
import MyNavbarComponent from './components/MyNavbar.ce.vue'

const vuetify = createVuetify({
    icons: {
        defaultSet: 'mdi',
        aliases,
        sets: {
          mdi,
        },
    },
})

customElements.define(
    'my-navbar',
    defineCustomElement(MyNavbarComponent, {
        plugins: [vuetify],
    })
)

src/defineCustomElementWithStyles.js

// defineCustomElementWithStyles.js
import { defineCustomElement as VueDefineCustomElement, h, createApp, getCurrentInstance } from 'vue'

export const defineCustomElement = (component, { plugins = [] } = {}) =>
  VueDefineCustomElement({
    styles: component.styles,
    render: () => h(component),
    setup() {
      const app = createApp()

      // install plugins
      plugins.forEach(app.use)

      const inst = getCurrentInstance()
      Object.assign(inst.appContext, app._context)
      Object.assign(inst.provides, app._context.provides)
    },
  })

src/components/MyNavbar.ce.vue

<template>
  <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/npm/vuetify@3/dist/vuetify.min.css" />
  <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/npm/@mdi/[email protected]/css/materialdesignicons.min.css" />
  <v-app>
    <v-app-bar>
      <template v-slot:prepend>
        <v-app-bar-nav-icon></v-app-bar-nav-icon>
      </template>

      <v-app-bar-title>App title</v-app-bar-title>

      <v-spacer></v-spacer>

      <v-btn icon>
        <v-icon>mdi-magnify</v-icon>
      </v-btn>

      <v-btn icon>
        <v-icon>mdi-heart</v-icon>
      </v-btn>

      <v-btn icon>
        <v-icon>mdi-dots-vertical</v-icon>
      </v-btn>
    </v-app-bar>
  </v-app>
</template>