I just attempted this mission today and can verify that Iris does not enter the car. Hopefully it gets fixed.
Interestingly, you can still finish it as if you retrieved her car by entering the vehicle and simply driving back to Dakota's. Iris will follow you on foot (she'll actually warp after you as you drive), and she'll still have a conversation with you about helping get her car as you drive to Dakota's. Once you talk to Dakota to finish the mission, Iris will send you a text, thanking you for helping her with her ride, and you'll get the objective to retrieve a bonus.
I do hope it gets fixed, but if you simply want to finish the quest with the objective "complete", you can do it in the manner I mentioned above.