How to limit user registration to a specific set of domains?

You can do it easily by writing a code inside your theme’s functions.php file. here is the code:

    function is_valid_email_domain($login, $email, $errors ){
 $valid_email_domains = array("gmail.com","yahoo.com");// whitelist email domain lists
 $valid = false;
 foreach( $valid_email_domains as $d ){
 $d_length = strlen( $d );
 $current_email_domain = strtolower( substr( $email, -($d_length), $d_length));
 if( $current_email_domain == strtolower($d) ){
 $valid = true;
 break;
 }
 }
 // if invalid, return error message
 if( $valid === false ){
 $errors->add('domain_whitelist_error',__( '<strong>ERROR</strong>: you can only register using @gmail.com or @yahoo.com emails' ));
 }
}
add_action('register_post', 'is_valid_email_domain',10,3 );

Source: https://www.eyeswift.com/allow-registration-from-certain-email-domain-wordpress/