How to Allow Non-Admin Users to Access WordPress Customizer. Check the latest version on the qSandbox - https://qsandbox.com/675

qs_blog675_grant_customizer_access.php

<?php

/**
 * How to Allow Non-Admin Users to Access WordPress Customizer. Published on the qSandbox - https://qsandbox.com/675
 * <a href="https://qsandbox.com/?utm_source=qs_blog_code&utm_medium=p675" title="Free Test/Staging WordPress site by qSandbox.com">Free Test/Staging WordPress site by qSandbox.com</a>
 * @param array $all_caps
 * @param array $cap
 * @param array $args
 * @return array array
 * @see https://codex.wordpress.org/Plugin_API/Filter_Reference/user_has_cap
 * @see https://wordpress.stackexchange.com/questions/12984/how-can-i-allow-the-editor-role-to-change-theme-settings
 * @see https://codex.wordpress.org/Class_Reference/WP_User#has_cap.28.24cap.29
 */
function qs_blog675_grant_customizer_access( $all_caps, $cap = [], $args = [] ) {
	remove_filter( 'user_has_cap', __METHOD__, 25, 3 ); // Prevent loop if other code calls this method.
	$cap = 'edit_others_posts';

	// Editor but without access to customizer
	if ( current_user_can( $cap ) ) {
		$all_caps['customize'] = true; // since WP 4.0
		$all_caps['edit_theme_options'] = true; // do we need this?
	}

	add_filter( 'user_has_cap', __METHOD__, 25, 3 ); // Re-add the call the action is consistent.

	return $all_caps;
}

add_filter( 'user_has_cap', 'qs_blog675_grant_customizer_access', 25, 3 );