Is the mother responsible for her daughter's death in a car accident after she allowed her to go alone to the shop near the house?

There is no blame upon you, Allah willing, if the child's going to the shop was normal, and no negligence or dereliction of duty originated from you, because merely leaving her in normal circumstances is permissible and not considered negligence. The scholars have stated that whoever does what is permissible for him, without transgression or negligence, and harm results from that, then there is no guarantee (liability) upon him. The owner of the car is responsible for her blood money, and his 'aqilah (male relatives on the father's side) are responsible for the expiation for accidental killing, as stated in the words of Allah Almighty: "Then [there must be] a compensation given to his family [of the deceased] and the freeing of a believing slave, but he who finds none, then [there is] a fast for two consecutive months as repentance to Allah. And Allah is ever Knowing and Wise."