Is it permissible for the father to buy the car promised in the vow, and pay its first installment, then for the vower to complete the remaining amount in installments, and would that fulfill the vow if the father only asked the vower to pay 5000 Riyals of the car's value?

The aforementioned vow is a permissible vow, and scholars have differed regarding its validity. The preponderant opinion, in our view, is that it is not valid, and thus nothing is binding upon you.

However, according to the view that it is valid, its fulfillment is not achieved except by purchasing the car. What you are doing by contributing to its price is not considered a purchase. You must either buy the car for your brother or offer expiation for breaking an oath.