Automatically update the display name to be set as the first/last name when the user is registered or the profile is saved or during checkout.

my-pmpro-set-user-display-name-as-first-last-name.php

<?php
/**
 * Automatically update the display name to be set as the first/last name when the user is registered or the profile is saved.
 *
 * You can add this recipe to your site by creating a custom plugin
 * or using the Code Snippets plugin available for free in the WordPress repository.
 * Read this companion article for step-by-step directions on either method.
 * https://www.paidmembershipspro.com/create-a-plugin-for-pmpro-customizations/
 */

add_action( 'user_register', 'my_pmpro_set_user_display_name_as_first_last_on_profile_save' );
add_action( 'profile_update', 'my_pmpro_set_user_display_name_as_first_last_on_profile_save' );

function my_pmpro_set_user_display_name_as_first_last_on_profile_save( $user_id ) {
	// Get the user object.
	$data = get_userdata( $user_id );

	// Set up the new display name based on the first/last name.
	$new_display_name = trim( $data->first_name . ' ' . $data->last_name );

	// Check whether the display name matches the login AND that the display name is not already the same as the first/last name.
	if (
		$data->display_name === $data->user_login
		&& '' !== $new_display_name
		&& $data->display_name !== $new_display_name
	) {
		// Update the display name to the first/last name.
		wp_update_user( [
			'ID'           => $user_id,
			'display_name' => $new_display_name,
		] );
	}
}

add_filter( 'pmpro_checkout_new_user_array', 'my_pmpro_set_user_display_name_as_first_last_on_checkout' );

function my_pmpro_set_user_display_name_as_first_last_on_checkout( $user_data ) {
	// Set up the new display name based on the first/last name.
	$new_display_name = trim( $user_data['first_name'] . ' ' . $user_data['last_name'] );

	if ( '' !== $new_display_name ) {
		$user_data['display_name'] = $new_display_name;
	}

	return $user_data;
}