May one leave a website store open over Shabbos?
Some Poskim rule one is not to allow his website business to remain open on Shabbos. Others rule it is permitted to leave one’s website open over Shabbos even if it performs sales of merchandise on Shabbos. Nevertheless, if it is common for also Jews who transgress Shabbos to purchase from the site, then it is proper to close it over Shabbos. There are various companies who provide Shabbos closing services for websites based on the customers current location.
Information service website: Those websites which provide information in exchange for a onetime fee [i.e. phone number; legal record; etc] and those websites who have an entrance fee for each time one enters, some Poskim rule are forbidden to remain open on Shabbos.
Third party sales: Those who sell products [not services] through a third-party platform, such as e-bay or Amazon.com, do not have to arrange with the platform to cease sales of their product over Shabbos.
 Shevet Halevi 10/57
 The reason: As aside for the questions of doing business and Sechar Shabbos, it ends up that his store is open 7 days a week and there is no greater belittlement of Shabbos than this.
 Piskeiy Teshuvos 252/2
 The reason: There is no obligation to shut down the site due it doing work for a Jew on Shabbos, as even actual vessels may do work for a Jew on Shabbos, if set up before Shabbos. [See Admur 252/1-2] There is also no prohibition of doing business on Shabbos being that the seller [the Jewish owner of the site] is not doing any physical action. [See Rav Akiva Eiger 59; Kesav Sofer 46; Maharam Shick 25; Minchas Yitzchak 3/34; Piskeiy Teshuvos 252 footnote 8] There is also no prohibition of Sechar Shabbos being that the buyer is receiving an item in return and the prohibition against Sechar Shabbos is only against paid labor, and only when the labor is done only on Shabbos. [See Noda Beyehuda 26 that a Jew may collect payment for wood of his that was used in the bathhouse; Minchas Yitzchak ibid; Piskeiy Teshuvos 252 footnote 9]
 Piskeiy Teshuvos 252/2; Nevertheless, this is not required from the letter of the law, as Lifnei Iver only applies when a Jew will for certain do the transgression. [See Piskeiy Teshuvos ibid footnote 6 and 245 footnote 55]
 Shevet Halevi 10/57; See Piskeiy Teshuvos 252 footnote 16
 The reason: As the prohibition of Sechar Shabbos applies in all cases that an actual item is not received by the buyer, but rather mere information. [Shevet Halevi ibid]
 The reason: As aside for all the other reasons mentioned for why there is no prohibition of Masa Umatan or Sechar Shabbos, it is also not one’s own website, and hence does not carry a belittlement of Shabbos.