In my story, fairies can't handle touching iron - they're allergic to it - but being near it is fine if they don't touch it. They wouldn't be able to handle a regular car very well, but could they handle a fiberglass car? Does it still have iron as well as fiberglass?
The primary components of most mass-produced cars are made of steel; plastics are generally limited to parts that don't bear a load (hood, trunk lid, fenders, bumpers, interior paneling, etc) while the chassis / unibody, engine, transmission, drivetrain, wheels, and just about everything else is still metal. Granted some of these parts are aluminum rather than steel, especially in sports cars, but the fact still remains that a car is riddled with metal, most of which is steel/iron. How close can your fairies safely get to iron? Can they safely handle it if they wear gloves? It would be virtually impossile to function in society if they can't get close to metals. Car keys are made of metal, too, unless they use a wireless remote. Once they sit inside, they'll find that the floor pans are metal, the seats are reinforced by metal, the steering column is metal... it goes on and on and on...
OK, come to think of it, I'm getting rid of that weakness. It doesn't really fit with my conception of my setting's fairies anyway. Never mind, then.
There have been a few that have no metal parts on the chassis at all. Nowadays in fact there are quite a few, but most of the mechanical parts are still metal, though ferrous metal parts can theoreticaly be kept to a tiny proportion of that. If is' just a atter that they cant touch iron, you're fine, but if they can't be within, say a meter of it, there are probably no cars in existence at the moment that fit the bill. however it is entirely possible to build a car of other metals entirely. The on;y thing i can think of in a car that absolutely has to be made of iron or steel are the piston rings, and even those could theoretically be made of something else if your fairies have plenty of money! Everything else can be made of aluminum, carbon fibre, brass, fibreglass, wood ( there is a car called the Marcos which used a lot of wood in the chassis until recently. the designer, Frank Costin, called it "God's plastic" ) and ceramic, which might be a possibility for the piston rings, in fact. f their car is electric though, it would have to have iron or rare earth.
