What programming language version was used to write the website, is it up to date and is it still being maintained and secure?
At the time of writing, PHP 7.1 is current with PHP 5 only receiving security updates until 31st December 2018. See the PHP lifetime plan here http://php.net/supported-versions.php
PHP 7 is being actively worked on giving developers new tools and significant performance improvements over previous versions. Your websites will likely require modification to work with PHP7.1 resulting in a good few hours work testing and fixing any issues.
Does your site use a package manager like Composer? Do the code dependencies require updating or are they still being actively maintained by the authors?
Some websites we see have bespoke features developed which are limited in usefulness or barely meet their design specification.
A common example are sites which manage their own mailing list and send bulk mail from the web server. There are far better solutions to manage and maintain a professional mailing list and distribution strategy, for instance, MailChimp along with all the many benefits it offers.
Use of such a service will improve the chance of delivery, provide useful reporting, automated unsubscribes, compliance with GDPR and better design tools to ensure your email looks great on every device.
Security and Hosting
Does the site meet modern security compliance standards?
Is the hosting suitable for the required uptime and performance based on its geographic audience and how mission critical availability is?
Your website hosting needs to be designed so it is responsive and fast for your audience, if you’re trying to sell products to people in Australia it’s no use hosting your website in London!
If you’re running a busy ecommerce store you’ll want to ensure that it is designed with no central points of failure and can be expanded to provide more resources at times of high demand.