This should go near the top of your PHP script on the page to listen for the submitted data.
<?php
if (isset($_POST['user_adresse'])) {
$userID = get_current_user_id();
$userADRESSE = $_POST['user_adresse'];
update_user_meta($userID , 'user_adresse' , $userADRESSE ); #'user_adresse' should be the name of the meta key
} ?>
and the form should be something like
<?php $user_adresse = get_user_meta(get_current_user_id(), 'user_adresse', true ); ?>
<form id="form_livraison" action="#formulaire" method='post'>
<input type="text" name="user_adresse" value="<?php echo $user_adresse;?>" />
<input type="submit" value="mettre à jour" name="submit_form" />
</form>
I would also recommend adding a nonce to the form for any front end user interactions.
Details can be found here under the section of Adding a nonce to a form
I have not tested this code so check for any misspellings etc!!
EDIT
<?php
if ( is_user_logged_in() )
{
$user_id = get_current_user_id();
$user_login = get_user_meta( $user_id, 'user_login', true );
$user_email = get_user_meta( $user_id, 'user_email', true );
$first_name = get_user_meta( $user_id, 'first_name', true );
$last_name = get_user_meta( $user_id, 'last_name', true );
$user_societe = get_user_meta( $user_id, 'user_societe', true );
$user_adresse = get_user_meta( $user_id, 'user_adresse', true );
$user_code_postal = get_user_meta( $user_id, 'user_code_postal', true );
$user_ville = get_user_meta( $user_id, 'user_ville', true );
$user_url = get_user_meta( $user_id, 'user_url', true );
echo $_POST['user_adresse']; // This will only show what is posted previously on the page - if you want to dsiplay the saved address use echo $user_adresse
if (isset($_POST['user_adresse'])) {
$userADRESSE = $_POST['user_adresse'];
update_user_meta($user_id , 'user_adresse' , $userADRESSE ); #'user_adresse' should be the name of the meta key
}
?>
<form id="form_livraison" action="#formulaire" method='post'>
<h2>Récapitulatif de vos coordonnées</h2>
<br />
<table style="width:320px">
<tr>
<td>Nom</td>
<td><?php echo $last_name; ?></td>
</tr>
<tr>
<td>Prénom</td>
<td><?php echo $first_name; ?></td>
</tr>
<tr>
<td>Adresse</td>
<td>
<?php if (isset($_POST['user_adresse'])) { ##This means that if you have just updated the post it will show what the post is updated to - otherwise the original meta is displayed.
echo $_POST['user_adresse']; } else { echo $user_adresse; }?></br><input type="text" name="user_adresse" placeholder="Ajouter/Changez votre adresse" /></td>
</tr>
<tr>
<td>Code postal</td>
<td><?php echo $user_code_postal; ?></td>
</tr>
<tr>
<td>Ville</td>
<td><?php echo $user_ville; ?></td>
</tr>
<tr>
<td>Web Adresse</td>
<td><input type="text" name="url" value="<?php echo $user_url;?>" /></td>
</tr>
</table>
<br />
<input type="submit" value="mettre à jour" name="submit_form" />
</form>