I've used Lansky's twice, the first time we stayed at the Anabella and our stop was at Von's, which was where I requested and I ASSUME they would be open early enough for a stop there if Target is closed).
The second time we stayed over off S Clementine and our stop was at Target, I requested Von's but he took us to Target, I assume bc it was closer to the hotel (there were a lot of issues with this driver, the grocery stop being the least of our issues, the other issues was his inability to install the provided car seats (which were also expired) and the lack of instructions for us to try to do it ourselves).
Anyway, if the Target is closed they should be able to provide a stop at an alternate grocery store.