There are two ways to acquire driver suits. You can either choose one as a level up reward or get one from a loot crate, the cheapest costing 70,000CR.
It is important to note that, unlike the car collection achievements, multiple copies of the same driver suit DO NOT count, since the game does not give you a second copy of the suit but gives you 5,000CR instead. Because of this, if you already have the car achievements, I recommend choosing the driver suits for your level up reward until you have the achievements for them.