Using AMP Script component for local storage.

amp-script-local-storage.php

<?php
/**
 * Local Storage Demo
 *
 * @requires AMP WordPress Plugin.
 */
add_action(
	'wp_head',
	function() {

		if ( ! is_amp_endpoint() ) {
			return;
		}

		/**
		 * layout="nodisplay" will not execute script also adding height="0" will not work.
		 */
		?>
		<amp-script layout="fixed" script="local-store-demo" height="1" width="1">
			<div id="site-name" style="display:none;">
				<?php echo esc_html( get_bloginfo( 'name' ) ); ?>
			</div>
			<div id="site-info" style="display:none;">
				<?php
				/**
				 * Retrives blog Info from predefined array.
				 *
				 * @return array blog info.
				 */
				function bloginfo_array() {
					$fields = array( 'name', 'description', 'wpurl', 'url', 'admin_email', 'charset', 'version', 'html_type', 'language' );
					$data   = array();
					foreach ( $fields as $field ) {
						$data[ $field ] = get_bloginfo( $field );
					}
					return $data;
				}
				echo wp_json_encode( bloginfo_array() );
				?>
			</div>
		</amp-script>
	<script id="local-store-demo" type="text/plain" target="amp-script">
		const testLocalStorage = () =>{
			const siteName = document.getElementById('site-name').textContent;
			const siteInfo = document.getElementById('site-info').textContent;
			localStorage.setItem('site-name', siteName);
			localStorage.setItem('site-info', siteInfo);
			console.log( localStorage.getItem('site-name') );
			console.log( localStorage.getItem('site-info') );
		}
		testLocalStorage();
	</script>
		<?php
	}
);
?>