WordPress basic AJAX example. https://help.codesnippets.pro/article/48-ajax-within-snippets

ajax.php

<?php

const ACTION_NAME = 'shea_ajax_example';

$handler = function () {
	check_ajax_referer( ACTION_NAME );

	wp_send_json_success( 'it works!' );
};

add_action( 'wp_ajax_' . ACTION_NAME, $handler );
add_action( 'wp_ajax_nopriv_' . ACTION_NAME, $handler );

add_shortcode( 'shea_ajax_button', function () {
	return '<button id="shea-ajax-button">Click me for AJAX</button>';
} );

add_action( 'wp_footer', function () {
	$data = [
		'ajaxurl' => admin_url( 'admin-ajax.php' ),
		'action'  => ACTION_NAME,
		'nonce'   => wp_create_nonce( ACTION_NAME ),
	];

	?>
	<script>
		const data = <?php echo wp_json_encode( $data ); ?>;
		const ajaxButton = document.getElementById('shea-ajax-button');

		ajaxButton.addEventListener('click', () => {
			const request = new XMLHttpRequest();
			request.open('POST', data.ajaxurl, true);
			request.setRequestHeader('Content-Type', 'application/x-www-form-urlencoded; charset=UTF-8');

			request.onload = () => {				
				if (200 > request.status || 400 <= request.status) {
					alert('AJAX failure :( We received error code ' + request.status);
					return;
				}
				
				const response = JSON.parse(request.responseText)

				alert('AJAX success! Message: ' + response.data);
			};

			request.send(`action=${data.action}&_ajax_nonce=${data.nonce}`);
		});

	</script>

<?php } );

Thank you for your explanation and the provided code. I would have hoped to find it earlier, as combining AJAX with a snippet is not that difficult after all.