Answer: The Shulchan Aruch (OC 307:4) allows one to give a non-Jewish person money before Shabbos for them to purchase something providing that they don’t specify that they should buy it on Shabbos. The Taz (OC 307:3) writes, however, that if one tells the non-Jewish person that they’re leaving on motzaei Shabbos, it is as if they specified that it must be purchased on Shabbos as there is no other realistic time for them to purchase it.
Following this, R’ Yitzchak Yaakov Weiss (Minchas Yitzchak 6:25) writes that one mustn’t book a taxi to be waiting for when Shabbos ends as inevitably, one is instructing the taxi driver to drive to their location on Shabbos. One would have to wait at least as long as it would take for the driver to arrive from the taxi rank or an average journey.
R’ Shalom Gelber and R’ Yitzchak Rubin (Orchos Shabbos 2:23:65), however, argue that the taxi driver’s journey to pick one up is incidental and not part of the instruction. They quote the Taz (OC 276:3) and Mishna Berura (276:27) who write that one is allowed to ask a non-Jewish person to wash their dishes even if that means that they will inevitably switch the lights on. Although they are doing so in order to perform something on your behalf, this is considered as doing so for themselves. Likewise, as the driver brings their car in order to perform their job, this is considered as if they are doing so for their own needs.
In conclusion, one is allowed to ask a non-Jewish driver to pick them up immediately after Shabbos even though they will be driving on Shabbos to get there.