Removing the website during one day will hurt SEO, but not as much. Depending on the traffic, some crawlers (the ones not scheduled to visit your site during Sabbath) won't even notice. But it will hurt for the users and for the others.
One solution is to set a server-side routine that changes the main page to a "closed for Sabbath" version, with the main controls disabled.
But your customer is going too far. From what was gathered in judaism.SE, there is little problem to leave automated systems run during the Sabbath, as long as they don't interact with the people observing Sabbath.
So a more sensible option is to leave the site running, and just disable automated notifications (email, sms, app) to the owners, and/or disable either the user login page or the shopping cart. This way users won't be able to "shop" during Sabbath.
But you can Tell your customer there is no problem on leaving the website on, even if it is E-commerce.
The Torah says you need to let your animals rest; what about your machinery? ("Shvisas keilim"). We follow Beis Hillel that it's not a problem, so your server can go on doing whatever it was doing.
There's also the problem of doing business on shabbos. A more complicated case is leaving my online store open on Shabbos (then after shabbos, you read the orders and ship them); Rabbi Heinemann shlit'a first prohibited this, but then reconsidered and allowed it. It's somewhat analogous to people putting envelopes in your mail slot; you deal with them after Shabbos.
From another question:
http://www.shemayisrael.com/parsha/halacha/Issue7.pdf mentions that one can keep a vending machine open on shabbos, because the pay one is getting is paid "Behavlaah" (Though I don't understand why, unless we're dealing with a contractor).
When one buys and sells through the internet, the transaction isn't finalized for a few days. Moreover, then it isn't "Schar Shabbos" because the work that one is getting paid for took place before Shabbos.
Credits to J for linking the questions in the comments above.