The prevalence of e-commerce poses a question regarding Shabbat. Must one must disable his retail website on Shabbat, in order to prevent transactions from occurring during Shabbat?
The Halachic precedent for this issue is found in the Poskim who discuss whether a Jew may keep his vending machines operating on Shabbat. Does the sale of goods and receipt of money on Shabbat by an automated machine present a problem on Shabbat? The responsa of the MaHarshag (Rav Shimon Greenfeld, 1860-1930, Hungary, Vol. 2:117) concludes that one may be lenient, assuming that the majority of customers are non-Jews, and the machines are not located on the Jew’s premises. This is also the opinion of the Be’er Moshe (Rav Moshe Stern of Debrecyn) in his Kuntres HaChashmal (p.84), Shemirat Shabbat K’hilchata (Rav Yehoshua Novert, 1927-1203, Hungary, Ch. 25:29) and Rav Shelomo Miller.
Presumably, the case of e-commerce websites is analogous to vending machines and would also be permitted on Shabbat.
The Poskim add that if one does keep his website open for business on Shabbat, it is preferable to adjust the settings so that the credit card details entered are processed only after Shabbat.
It should be noted that some Poskim reject this leniency and prohibit e-commerce on Shabbat. One who is stringent in accordance with their opinon, Tavo Alav Beracha, is considered praiseworthy. For a beautiful and clear analysis of this subject, see "Commerce on Shabbat,"(p. 184).
SUMMARY: One may keep his e-commerce website operating on Shabbat.