Automatic processes that don't require human intervention are generally permitted. There are some exceptions for things that violate the atmosphere of Shabbos.
However, transactions may have a specific prohibition which applies even if they are automatic.
The Rambam (Hil. Shabbos 23:12) is clear that the buyer and seller are both included in the prohibition of business on Shabbos:
וכן אסור לקנות ולמכור, ולשכור ולהשכיר--שמא יכתוב
It is prohibited to buy and sell ... lest one come to write
The Divrei Malkiel 4:2 explains the prohibition of the seller:
דהא עכ"פ צריך להראות לו היכן מונח הדבר וגם משגיח כמה הוא נוטל
The seller must show him where the item is, and oversee how much he takes.
According to this, it would seemingly be OK by ecommerce, where the seller is completely uninvolved in the actual event of the transaction.
The Chelkas Yaakov 1:68 explains it as follows:
האיסור למוכר משום הריצוי והסכם ג"כ
This would hypothetically include the sales that happen on Shabbos with the owner's consent, however the Chelkas Yaakov goes on to explain (by vending machines) that the seller's willingness and consent to the sale are really present from before Shabbos, and therefore nothing wrong is being done on Shabbos.
However, the Eizer Lashovsim Siman 10 explains it as follows:
האיסור הוא להיות חלק ממקו"מ בשבת. וזהו גופא מה שגזרו חז"ל שלא יהא חפצים יוצאים מרשותינו או נכנסים לרשותינו בשבת ע"י מקו"מ ... דהאיסור המציאות ולא המעשה
The prohibition is to be party to a sale on Shabbos, and this is exactly what Chazal were decreeing, that there should not be objects changing ownership on Shabbos through commerce ... the prohibition is the reality of the transaction and not the act.
This would include a sale even if you did not actively participate or even have active awareness of its happening.
