In an article in the journal Techumin (link), R. Shlomo Dichovsky addressed the permissibility of keeping open on Shabbos a website that sells items. He discusses four possible halakhic issues before reaching a conclusion:
1. Assisting a sinner – By leaving the website open, you are assisting someone in purchasing an item thewreby violating Shabbos. R. Dichovsky quotes R. Shlomo Zalman Auerbach (cited in Shemiras Shabbos Ke-Hilkhasah, ch. 20 n. 70) as permitting leaving out soda machines on Shabbos because the machines are not inherently forbidden. You can leave them out for weekday purpose and anyone who uses them on Shabbos is commiting the violation on their own. You also aren’t doing anything yourself on Shabbos. However, you should place a notice on the website advising customers that all transactions are completed after Shabbos.
2. Earning Money on Shabbos – The Noda Bi-Yehudah (Vol. 2, Orach Chaim no. 26) rules that it is permissible for women to use a mikveh on Shabbos and pay afterwards for the wood that was burned to heat the water rather than the service on Shabbos. Similarly, suggests R. Dichovsky, customers of a website pay after Shabbos for an item and not a service they received on Shabbos (but what about websites that sell a web service?).
3. Maris Ayin – A Jew is not allowed to have a Gentile work on on his property on Shabbos because it gives the mistaken impression that the Jew is doing the work (Shulchan Arukh, Orach Chaim 243:1). However, in the case of the website the work is automatic and there is no reason to think that the Jew or one of his employees is doing the work. This has additional force when the website has a notice that all transactions are completed after Shabbos (or all weekend transactions are completed on Monday).
4. Denigrating Shabbos – There is a general concern among halakhic decisors that automatic work will detract from the spirit of Shabbos. There are no strict guidelines for what falls into this category but some acts have been forbidden for this reason. R. Dichovsky argues that this concern is inapplicable because the websites function entirely outside of the Shabbos sphere. It cannot detract from or denigrate Shabbos when it is not a part of the Shabbos experience.
Therefore, R. Dichovsky concludes that he cannot forbid keeping website open for sales on Shabbos but there is a negative “taste” to the practice.