functions.php

<?php
/**
 * Plugin Name: WPSE 282139
 * Plugin URI: https://wordpress.stackexchange.com/questions/282139/how-to-make-get-theme-mod-work-with-ajax-in-the-customizer-preview
 */

add_action( 'wp_ajax_my_request', function () {
	ob_start();

	echo get_theme_mod( 'position', 'top' );

	wp_send_json_success( ob_get_clean() );

	wp_die();
} );

add_action( 'wp_enqueue_scripts', function() {
	wp_enqueue_script(
		'my-request',
		plugin_dir_url( __FILE__ ) . 'request.js',
		array( 'jquery' ),
		null,
		true
	);

	wp_localize_script(
		'my-request',
		'myArgs',
		array(
			'url'    => admin_url( 'admin-ajax.php' ),
			'action' => 'my_request',
		)
	);
} );

add_action( 'customize_register', function( WP_Customize_Manager $wp_customize ) {
	$wp_customize->add_section( 'position', array(
		'title'    => 'Position',
		'priority' => 1,
	) );

	$wp_customize->add_setting( 'position', array(
		'default' => 'top',
	) );

	$wp_customize->add_control( 'position', array(
		'label'   => 'position',
		'section' => 'position',
		'type'    => 'select',
		'choices' => array(
			'top'    => 'Top',
			'right'  => 'Right',
			'bottom' => 'Bottom',
			'left'   => 'Left',
		),
	) );
} );

request.js

/* global myArgs, console, _wpCustomizeSettings */
(function( jQuery ) {

	'use strict';

	var data = {
		action: myArgs.action
	};

	function request() {
		var url = myArgs.url, urlParser;

		// Workaround defect introduced in WordPress 4.8.2 where preview nonce is not included in request.
		if ( 'undefined' !== typeof _wpCustomizeSettings ) {
			urlParser = document.createElement( 'a' );
			urlParser.href = url;
			urlParser.search += '&customize_preview_nonce=' + _wpCustomizeSettings.nonce.preview;
			url = urlParser.href;
		}

		jQuery.post( url, data, function( response ) {
			console.log( response );
		} );
	}

	document.addEventListener( 'DOMContentLoaded', function() {
		request();
	} );

})( jQuery );