In this situation, you don’t need to ask the writer for permission. You do, however, need to clearly state the source in the article. Some authors will use the customer’s name as it appears in their review and the name of the website, but it is more common to remain general: “According to a customer reviewer at BookSellers . . .” If the reviewer is someone who is well known, then you may want to use the name to make that context clear.
It is important to know that customer review sections at retail websites are often open forums, where anyone can post comments anonymously. You often don’t know who is behind the review and what their motivation is for giving it. A consensus among numerous reviews can give a general idea of how the book was received, but individual reviews don’t hold much credibility. Keep that in mind when choosing when and what to quote from such sources.