Is the prayer valid for someone who prayed sitting in a car facing the Qiblah, when there was a prayer rug available in the banquet hall that he did not take out of shyness?

Standing is a pillar of prayer, and your prayer performed sitting in the car is not valid and must be repeated. The absence of a prayer mat does not justify this, for the entire earth is a mosque and a means of purification, as stated in the authentic Hadith (definition): "The earth has been made for me a place of prostration and a means of purification, so any man of my Ummah, wherever the time of prayer overtakes him, should pray."