"The owner undertakes appropriate technical and organizational measures to protect personal data against loss or any form of unlawful processing. These measures, taking into account the state of the technology and the cost of implementation, provide an adequate level of security in view of the risks involving the processing and nature of data to be protected. "
Most data that you store as a site owner, you don't want ending up in the hands of someone else. That applies to all written (SEO) texts to product descriptions and images. In addition, your site also collects data from visitors and customers. In addition to protecting this data from loss (backups!), you must also adequately protect this data against abuse and leaks. Due to software errors and / or hackers, this data can be obtained by others, which should be avoided as much as possible. There are legal requirements for this security and these differ between the types of data.
These include: Name, Address, Zipcode / City and Date of birth. If your store uses accounts (login option), the passwords are also personal: These are also a popular target for hackers so they must be sufficiently secured (hashed). Also check if there are no logs where (accidentally) these passwords are stored in original form (un-encrypted). Some recent leaks involved these kind of 'raw logs' which were exposed. Ask your developer about it!
Most e-commerce sites will use a payment provider to handle payments. Usually, the consumer first selects a payment method and then proceeds to a website of this payment provider where the data is entered. I think it's the best choice not to store bank account numbers or similar data yourself: it's usually not necessary and very sensitive information. The same goes for Credit Card data. And make sure that your credentials with your payment provider (Mollie, Targetpay, Internetbox, etc.) are protected; use 2-factor-authentication where possible.
Often you have a newsletter and manage a newsletter list of recipients. Besides that you can subscribe your clients as well (after permission!). Make sure that this data, along with your login details, remain secure. Email addresses are a popular prey for spammers and are also characterized as 'personal data'.
Some aspects of web store data security.
The physical security takes place in the data center where your site is located. With almost all major hosting providers around the world, this security quite high. Only authorized people can reach the servers at the datacenter. Because of the many online hack possibilities, this form of unauthorized (physical) access is not really a problem if you run a website without extra high risk. (It would only be an issue for banks, payment service providers and really big corporations)
If you develop your website (think of WooCommerce, Magento, etc.), the developer often has access to the database. For many systems, this access is necessary or useful during development and installation (for example, via phpMyAdmin software). Make sure this access is well protected and removed/blocked after you go into production. If access is no longer required: delete the (development) users.
Your employees or fulfilment agencies will need to be able to log in to the back office for processing orders. Create accounts with as few rights as possible: so that users can only see the data that is strictly necessary for their role. Provide new passwords every now and then, and block employee accounts after the employee has left (don not use generic accounts like 'report', 'shipment' and share them). Only use personal accounts (no company-wide account), so users can block and can often see what someone else has done. In Europe (GDPR) it is also required that from every change which is made in personal data, the employee, time, date and change is logged.
The biggest problem with data leaks is the software used. For an open source package, there are many ways to keep the software up-to-date. Therefore, always check the updates of the software you are using. And that's not just "Wordpress" itself but also all plugins, themes, and server software (apache, mysql, phpmyadmin etc.). If you don't have enough skills for this, you cannot provide adequate security and you need to hire a specialist to do so.
In case of a business sale (a whole company changes shareholders), all of the above data remains within the company, and requires little contractual effort. However, for example, it is important for buyers to be sure which (part of a) address file gives permission for email (opt-in).
If you acquire a web store through an asset transaction, then transferring becomes more complicated. The customer information (including personal information), opt-in addresses etc. must be transferred from one company to another. The former owner must then carefully remove this information from his company (except minimal required info for the tax authorities).