I would like to purchase an item on eBay but the auction isn’t slated to finish until Friday night. Would I be allowed to place an automated bid on Friday prior to shabbat if the auction won’t finish until after shabbos has begun?
Yes, this is fine. The fact that the sale will be completed automatically on Shabbos does not constitute a problem. The reason for this is that there is no desire that the sale should be completed specifically on Shabbos; if this happens, no prohibition is involved.
See Noda Biyhuda (Tinyana, Orach Chaim 26), who writes that there is certainly no problem for somebody to receive money for a sale that is performed on Shabbos.
The sale itself is usually prohibited, but in cases where it is permitted (see Orach Chaim 323:4 concerning purchases for the purpose of Shabbos, and Orach Chaim 245:5 concerning a non-Jew who sells on behalf of a Jew), there is no prohibition of “making money” on Shabbos. Just as it is permitted, in this instance, to “make money,” it is likewise permitted for the auction to be completed.
A similar idea is found in Shut Beis Shlomo (Orach Chaim 36) and in the Tehilla Le-David (244:11).
Based on this principle, authorities have permitted the placement of slot machines (selling drinks or other commodities), even when they will be used on Shabbos (in areas where the use on Shabbos will be by non-Jews; see Shut Maharshag 2:117; Shemiras Shabbos Kehilchasa, in the name of Rav Shlomo Zalman Auerbach, Chap. 29, note 71).
Note that where a specific instruction to transfer the money on Shabbos must be given, Iggros Moshe (Orach Chaim 3:44) prohibits (concerning the sale of stocks on Shabbos). In this case, there is no specific instruction referring to Shabbos, so that there is no prohibition.