I see you're using VueJS. This makes something like this easy!

How I would do it is as follows:

add the following to your data() : isClicked: false

wrap your code into a container like this:

<div v-if='isClicked'>
    <form action="#" id="Reactie_form">
        <input id="Form_voornaam" type="text" placeholder="Voornaam" style="width: 50%"> <br>
        <input id="Form_achternaam" type="text" placeholder="Achternaam" style="width: 50%"><br>
        <input id="Form_email" type="email" placeholder="E-mail" style="width: 50%">
        <textarea id="Form_bericht" style="width: 100%"></textarea>
        <input id="files" type="file" @change="fileSelected" multiple accept=".png, .jpg, .webp, jpeg, .gif">
        <input type="button" class="sendButt" value="Verstuur" @click="submitReactie">
    </form>
    <span v-if="!files || !files.length"></span>
    <ul v-else>
        <li v-for="files in files" :key="files.name">{{ files.name }}</li>
    </ul>
</div>

and at the following to your button: <button @click='isClicked = true'>

Wrap your code and add the wrapper container a hidding class. then you can fetch the click event and toggle the css by javascript.