User Without Email?

It's possible, just not so easily via the Users page without some hackery.

The WordPress Function wp_create_user will let you insert users without email addresses, so a little custom plugin to accept a login name and password, and a call to wp_insert_user or wp_update_user should work for you. Unfortunately I don't have time to code it up for you, but perhaps this will point you in a direction.

$userdata = array ('user_login' => 'someuser', 'user_pass' => 'swordfish');
$new_user_id = wp_update_user( $userdata );

Update for people looking for a solution with the REST-API: Unfortunately, this is not possible by the rest-api. the email is required there. see: https://developer.wordpress.org/rest-api/reference/users/#create-a-user

I find a way on youtube , when I search "how to create users without email in wordpress" !

1. install "code snippets" plugin
2. choose "add new" in this plugin
3. choose a name for that. for example "Email not required code"

4.Enter this code:

// This will suppress empty email errors when submitting the user form
add_action('user profile_update_errors' , 'my_user_profile_update_errors', 10, 3);
function my_user_profile_update_errors($errors, $$update, $user){
$errors->remove('empty_email');
}

// This will remove javascript required validation for email input
// It will remove the '(required)' textin the label
// Work for new user , user profile and edit user forms
add_action('user_new_form', 'my_user_new_form', 10, 1);
add_action('show_user_profile', 'my_user_new_form', 10, 1);
add_action('edit_user_profile', 'my_user_new_form', 10, 1);
function my_user_new_form($form_type) {
?>
<script type= "text/javascript">
jQuary('#email').closest('tr').removeClass('form-required').find('.description').remove();
// Uncheck send new user email option by default
<?php if (isset($form_type) $$ $form_type === 'add-new-user') : ?>
jQuery '#send_user_notification').removeAttr('checked');
<?php endif; ?>
</script>
<?php
}

5. select "only run in administrayion area and choose "save changes and activate
6. now check , it works!

Using Wordpress 4.7.2, I solved this with the following hooks:

// This will suppress empty email errors when submitting the user form
add_action('user_profile_update_errors', 'my_user_profile_update_errors', 10, 3 );
function my_user_profile_update_errors($errors, $update, $user) {
    $errors->remove('empty_email');
}

// This will remove javascript required validation for email input
// It will also remove the '(required)' text in the label
// Works for new user, user profile and edit user forms
add_action('user_new_form', 'my_user_new_form', 10, 1);
add_action('show_user_profile', 'my_user_new_form', 10, 1);
add_action('edit_user_profile', 'my_user_new_form', 10, 1);
function my_user_new_form($form_type) {
    ?>
    <script type="text/javascript">
        jQuery('#email').closest('tr').removeClass('form-required').find('.description').remove();
        // Uncheck send new user email option by default
        <?php if (isset($form_type) && $form_type === 'add-new-user') : ?>
            jQuery('#send_user_notification').removeAttr('checked');
        <?php endif; ?>
    </script>
    <?php
}

The "Import Users from CSV" plugin will allow you to create users with no email address. I added them with...

last_name,first_name,display_name,user_login
Gamble,Bill,Bill Gamble,BillGamble

Our club leverages the Wordpress wp_users database, but we need to have on file even those members who have no email

This is a pretty old question, but is still relevant.

Anyway, @Milo definitely had the right idea.

Here is what I use and it works great. This is very convenient when you want to create users, but prefer to keep your wordpress backend on lockdown. For example if you have a blog and prefer your content authors submit their work to you, this enables you create a user for them while not actually giving them access. Or if you prefer to blog under an alias. Or it's even useful for testing purposes or I guess those "sketchy" people who want to make up a bunch of fake users for their site (you know who you are).

Add this to functions.php temporarily or use it with a useful plugin like code snippets.

function wpse_22754_insert_new_user() {

    $user_data = array(
    'ID' => '', // automatically created
    'user_pass' => 'swordfish',
    'user_login' => 'someuser',
    'user_nicename' => 'Some User',
    'user_email' => '',
    'display_name' => 'John',
    'nickname' => 'jsmith',
    'first_name' => 'John',
    'last_name' => 'John',
    'user_url' => '',
    'user_registered' => '2015-06-26 10:55:55', // leave empty for auto
    'role' => get_option('default_role') // administrator, editor, subscriber, author, etc
    );

    $user_id = wp_insert_user( $user_data );

    }
add_action( 'admin_init', 'wpse_22754_insert_new_user' );

If you are looking to update users from the backend after creating them, add this and you'll avoid getting the error/warning notification.

function wpse_22754_empty_email_error( $arg ) {
    if ( !empty( $arg->errors['empty_email'] ) ) unset( $arg->errors['empty_email'] );
}
add_action( 'user_profile_update_errors', 'wpse_22754_empty_email_error' );

If you'd rather use a plugin this one will do the job most likely:

https://wordpress.org/plugins/allow-multiple-accounts/

According to WordPress' documentation, wp_create_user also let's you insert users without e-mail, as it is an optional parameter. All you need to do is provide a unique username and a password.