You have to require('datepicker') in .vue component.

I'll give you example here, i just download datepicker plugin from github & let me know if this one is you used there.

I used bootstrap.js, dashboard.js (cant be anything for you), and 'datepicker.min.js' where i put in resources/assets/js/ folder & stylesheet files dashboard.scss in resources/assets/sass/ folder.

bootstrap.js

/**
 * We'll load jQuery and the Bootstrap jQuery plugin which provides support
 * for JavaScript based Bootstrap features such as modals and tabs. This
 * code may be modified to fit the specific needs of your application.
 */
window.$ = window.jQuery = require('jquery');

dashboard.js

const Vue = require('vue')

Vue.component('example', require('./../components/Example.vue'))
new Vue({
    el: '#app'
});

Example.vue

<template>
    <div class="col mt-5">

        <div class="row" style="margin-top: 200px">
            <input id="date">
        </div>
    </div>
</template>

<script>
require('./../datepicker.min'); // depend on your folder structure;
    export default {
        mounted() {
            $('#date').datepicker()
        }
    }
</script>

dashboard.scss

@import "../datepicker.min.css"; // depend on your folder structure

Compile the asset with nm run dev or npm run prod. Then include the script with order app.js & then dashboard.js and of course the stylesheet.

And then voila, we gate datepicker on the page. Here is the screenshot

Hope this will work for you.