Is it permissible to continue paying installments for a car to a company that deals with usury, after purchasing it?

If you mean selling the car before paying off its installments, then it is not permissible, because it is selling what one does not own. However, if you will replace the lessee in the lease-to-own system, then it is permissible, provided that the system adheres to the Sharia guidelines and the company's usurious dealings are separate from your dealings with it.