I have a form where user insert their names, once they submit the form, the page should check if the name provided already exists, if so, should return with an error, other wise
I got it from your comments that you have the following:
$_POST["user-submitted-name"]; // name from the form
$names = get_meta_values('user_submit_name'); // array with names not allowed
Then it looks like the easy way:
if (in_array($_POST["user-submitted-name"], $names)) {
// posting not allowed
echo '- Name already used';
} else {
// posting is allowed
// save posting here...
// and then go to success page:
header("Location: success.html"); // redirect
exit(); // and stop here
}