Answer: The rishonim give different reasons for why one mustn’t conduct business transactions on Shabbos. Rambam (Shabbos 23:12) writes that we are concerned that one may record the transaction. Rashi (Beitza 37a) adds that there is a prohibition against discussing business matters on Shabbos.
The Gemara (Shabbos 18a) writes that one is allowed to open a flow of water before Shabbos that will run onto a garden on Shabbos. Thus, the Shulchan Aruch (OC 252:5) writes that one is allowed to have one’s utensils perform work for them on Shabbos. There is a machlokes as to whether this applies to business, too, however.
R’ Akiva Eiger (Teshuvos 1:159) writes that one cannot set up a business transaction before Shabbos that will be performed on Shabbos. R’ Avraham Borenstein (Avnei Nezer OC:51) explains why scheduled business transactions are different to other automated melachos.
R’ Meir Simcha of Dvinsk (Ohr Sameach, Shabbos 23:12) maintains, however, that there is no difference between business and other melachos. Likewise, the Maharam Shik (131) writes that one may submit a bid before Shabbos for an auction that will end on Shabbos. Although the auctioneer announces the final bid, by submitting the bid in advance, one has set the ball rolling. Additionally, while winning the actual bid obligates the winner to pay, the transaction is not complete until they have paid (See Teshuvos Vehanhagos 1:232).
In conclusion, one may bid for an item on eBay that ends on Shabbos.