I'm about to get rid of my old car and get a new one. Does Islam permit his followers to give the car to the organizations like Salvation Army for example?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

If this organization to which you want to offer a car deals with any forbidden activity in religion like supporting Christian missionaries or financially helping what is Islamicly forbidden then it is not permissible to offer them any kind of help or assistance, because if you do so, you are helping them in sin and transgression. Allaah says (which means): {Help you one another in virtue, righteousness and piety but do not help one another in sin and transgression. And fear Allaah. Verily, Allaah is Severe in punishment.}[5:2]. 

Rather, we advise you to donate to Islamic trustworthy organizations which help the poor, needy and widowed Muslims all over the world. These are the people who are most in need of help. Or, you can donate it or sell it and give the money to mosques, Islamic centres, or schools which are teaching the Qur'an and so on, as in doing so, you are helping in virtue and righteousness.   

Allaah knows best.