I don't know much about cars, and correct me if I'm wrong, but I think that if the tire gets too hot, then it can explode. I'd almost say that if it was true that there are animal by products in tires to make them cooler and that there was no other alternative, that this is almost a necessary evil to save many lives, lots of pain and suffering, and millions of dollars. Not very vegan sounding, but good tires save lives.
Let's just hope that the tires you have on your car, vegan or not, are safe. I just took one of those defensive driving classes, and one of the things the teacher stressed was the importance of good tires on your car. I dont remember all the reasons (that's "boy stuff", haha), but I do remember its really important to have.
Plus, with how honest some of the workers at the local garages are, I would bet that a lot of them would say that the most expensive tires they have are the ones that just happen to not have any animal by products in them. (Haha, plus try explaining to some of them, or at least the ones I've seen and gone to, what being vegan is and everything....Not trying to be stereotypical or anything, I'm just thinking of the expereinces I had at a couple garages getting my car fixed lately.) I think it would be too difficult to get the tire ingredients and probably not worth your time, effort, and frustration.