There are different views on the subject. Some say (see below) that you are doing business on Shabbat and therefore prohibited. Others say that you not making a sale, just taking orders – akin to receiving a mail order.
In any event, you should talk to a posek.
From Rabbi Mansour’s daily halacha site:
“Rav Shemuel Pinchasi (contemporary) addresses this question in his new book (listen to audio for precise citation) and rules that it is forbidden to allow the site to run on shabbat. Even if the majority of those accessing the site are not Jewish, and even though the site owner will process the orders only after shabbat, he must have the site shut down during shabbat. Rav Pinchasi cites a number of leading Halachic authorities who hold this view, including Rav Yosef Shalom Elyashiv, Rav Chayim Kanievsky, and Rav Yonah Metzger.”