手順1

https://github.com/constancecchen/object-fit-polyfill/blob/master/dist/objectFitPolyfill.min.js
ここでjsをダウンロードして、static/assets/js/object-fit-polyfill.min.jsに配置

手順2

nuxt.config.jsでjsファイルを読み込ませる

nuxt.config.js

export default {
  head: {
    script: [
      { src: '/assets/js/object-fit-polyfill.min.js', body: true }
    ]
  }
}

手順3

ビデオコンポーネントの作成

components/atoms/video.vue

<template lang="pug">
.a-video(:class="{'is-cover': cover}")
  video.a-video-movie(v-bind="bindMovie")
</template>

<script>
export default {
  props: {
    src: { type: String, required: true },
    poster: { type: String, required: false, default: '' },
    autoplay: { type: Boolean, required: false, default: false },
    loop: { type: Boolean, required: false, default: false },
    playsinline: { type: Boolean, required: false, default: false },
    muted: { type: Boolean, required: false, default: false },
    ext: { type: String, required: false, default: 'mp4' },
    res: { type: Boolean, required: false, default: false },
    cover: { type: Boolean, required: false, default: false }
  },
  computed: {
    bindMovie() {
      return {
        src: this.srcset(),
        poster: this.poster,
        autoplay: this.autoplay,
        loop: this.loop,
        playsinline: this.playsinline,
        muted: this.muted,
        type: 'video/mp4',
        'data-object-fit': this.cover ? 'cover' : false
      }
    }
  },
  methods: {
    srcset() {
      const res = this.$store.getters.innerWidth < 768 ? '_sp' : '_pc'
      return `${this.src}${this.res ? res : ''}.${this.ext}`
    }
  }
}
</script>

<style lang="stylus" scoped>
.a-video
  line-height 0

  &.is-cover
    position relative
    overflow hidden
    width 100%
    height 100%

    .a-video-movie
      position absolute
      width 100%
      height 100%
      top 0
      left 0
      object-fit cover
</style>

あと、初期化時やリサイズ時、良きタイミングでこれ使う

objectFitPolyfill()

mounted() {
  objectFitPolyfill()
  window.addEventListener('resize', objectFitPolyfill())
}

参照

object-fit-polyfill

Nuxt.jsのそのほか

Nuxt.jsを使う