Install svg-inline-loader for webpack :

npm install svg-inline-loader --save-dev

Add it to your loaders in module.loaders section of webpack.config.js:

{
  test: /\.svg$/,
  loader: 'svg-inline-loader'
}

And now the tricky thing is to modify your vue-loader options to use require for vector:src attribute which will in turn use the svg-inline-loader:

transformToRequire: {
  // use for vector:src
  vector: 'src',
  // img:src & image:xlink:href url transform (vue default)
  // https://github.com/vuejs/vue-loader/blob/master/docs/en/options.md
  img: 'src', image: 'xlink:href'
}

(optional) Use vector as a top-level component:

import Vector from './Vector'
Vue.component('vector', Vector)

And now in your .vue files you can use it like "traditional" images :

<template>
  <div>
    <vector src="~assets/my-image.svg"></vector>
  </div>
</template>