Instagram API integration tutorial, as discussed here: http://zoerooney.com/blog/tutorials/custom-instagram-feed-stats-integration-high-traffic/

instagram.php

<div id="instagram" class="widget">
	<?php

	// Set variables
	$image_number = 2;
	$image_size = 'low_resolution'; //options: low_resolution (306), thumbnail (150), standard_resolution (612)

	// Create the API URL for the user data
	$user_access_token = '123456.xxxxxxxx.zzzzzzzzzzzzzzzzzzzzzzz';
	$user_data_url = 'https://api.instagram.com/v1/users/self/?access_token=' . $user_access_token;

	// Use a transient for the feed URL (performance boost)
	if ( false === ( $user_data = get_transient( 'instagram_user_data' ) ) ) {
		$user_data = @file_get_contents( $user_data_url );

		// Put the results in a transient. Expire after 4 hours.
		set_transient( 'instagram_user_data', $user_data, 4 * HOUR_IN_SECONDS );
	}

	// Decode the JSON in the file
	$user_stats = json_decode( $user_data, true );
	$user_image_count = $user_stats['data']['counts']['media'];
	$user_followers_count = $user_stats['data']['counts']['followed_by'];
	$user_follows_count = $user_stats['data']['counts']['follows'];

	// Create the API URL for the user feed
	$user_feed_url = 'https://api.instagram.com/v1/users/self/media/recent/?access_token=' . $user_access_token . '&COUNT=' . $image_number;

	// Use a transient for the feed URL (performance boost)
	if ( false === ( $user_feed = get_transient( 'instagram_user_feed' ) ) ) {
		$user_feed = @file_get_contents( $user_feed_url );

		// Put the results in a transient. Expire after 4 hours.
		set_transient( 'instagram_user_feed', $user_feed, 4 * HOUR_IN_SECONDS );
	}

	// Decode the JSON in the file
	$user_images = json_decode( $user_feed, true );

	?>
	<div class="widget-content">
		<h3 class="widget-title">FOLLOW ALONG ON INSTAGRAM</h3>
		<div id="instagram-feed">
			<?php $image = 0;
			// Loop through images in the feed
			for ( $user_image = 0; $user_image < $image_number; $user_image++ ) {
					$user_image_data = $user_images['data'][$image];

					// Display the images:
					$user_link = $user_image_data['link'];
					$user_caption = $user_image_data['caption']['text'];
					$user_image_url = $user_image_data['images'][$image_size]['url'];
					?>
					<div class="instagram-image">
						<a href="<?php echo $user_link; ?>" target="_blank"><img src="<?php echo $user_image_url; ?>" alt="<?php echo $user_caption; ?>" /></a>
					</div>
				<?php 
				$image++;
			} ?>
		</div>
		<div class="instagram-info">
			<a href="http://instagram.com/username" target="_blank">@username</a><br>
			<?php echo $user_followers_count; ?> followers<br>
			<?php echo $user_image_count; ?> posts<br>
			<?php echo $user_follows_count; ?> following
		</div>
	</div>
</div>

Tutorial at: https://zoerooney.com/blog/tutorials/custom-instagram-feed-stats-integration-high-traffic/