Component com formulário de exemplo para upload de arquivo com vuejs 2: https://portal.code.education/admin/curso/duvidas/5246/view

Upload.vue

<template>
    <div class="container">
        <form method="POST" @submit.prevent="send()">
            <label>Arquivo</label>
            <input @change="setFile" type="file">
            <button type="submit" class="btn">Enviar</button>
        </form>
    </div>
</template>
<script type="text/javascript">
    export default {
        data() {
            return {
                file: null
            }
        },
        methods: {
            setFile(e) {
                var files = e.target.files;
                this.file = files[0];
            },
            send() {
                let data = new FormData();

                data.append('file', this.file);

                this.$http.post('teste', data, {
                    headers: {'Content-type': 'multipart/form-data'}
                });
            }
        }
    }
</script>