Depends how much you want the writing to matter. If you just want to have some choices dictated by dice, a simple d6 system like what "Age of Sigmar: Soulbound" or "Shadow Run" use will work fine. This is more or less the dice pool type of systems Yankee already covered. As the GM, you can look at every development during a race and determine how difficult it should be to overcome. Maybe make yourself some charts to determine how difficult some common tasks are. Then make a more general one to give everyone an idea what to expect. Maybe an easy task takes just one success, while mid ones take three, hard is five, and nigh impossible is seven or higher. These numbers are going to depend on a lot of other factors to work right. AOS:S generally considers a roll of four or higher to be a success while SR requires a five. SR also has you rolling more dice most of the time. There are some other cool gimmicks you can use too. Like in SR, if half of your dice turn up ones, you "glitch" which is a sort of mixed success where there's a complication. Maybe you successfully shove a car off the road, but you get your fender stuck in their doorjam and it gets ripped off. Because glitches are tied to the number of dice, you're less likely to glitch if you have a huge dice pool. 

If you want something more complicated, systems exist. Just be aware that as you add more rules, the role play will start to become more of a game. Players will become more competitive and will likely be motivated to perform well over making interesting developments. Maybe a game is what you're looking for, in which case I would go for more complex rules.