The majority of vehicle design need not change. This is because the mass of the vehicle remains the same, no matter what the gravity is. When dealing with the classic equation:
$$F = mA$$
The effect of gravity is in the "A" (acceleration) term. When the gravitic force is perpendicular to the vehicle (when you're cruising down main trolling for a date Friday night), that force has no affect on acceleration. On the other hand, when you're ascending or descending a hill, a component of gravity is affecting your acceleration. But, in the end, it's just force, so there's little difference from modern design. But, of these differences, I can think of the following:
Fuel and coolant pressures will change moderately between gravities. However, this is easily overcome with a pressure regulator that consistently feeds the pressure appropriate for the engine despite changes in torque. This problem is already solved today (since torque and fuel feed already change when you climb a hill) using the engine's vacuum system. Basically, as vacuum pressure increases, so does fuel pressure or volume (depending on how the engine is actually designed).
Braking will change between the gravities, but again, only somewhat. The greatest problem will be the need for increased braking descending a hill on a high-G planet. Once again, modern tech already exists (via the ABS systems) to adjust both braking pressure and oscillation frequency to manage braking heat vs. braking friction. In the end, you'll use the same pads on both planets — they'll just last longer on the low-G world.
Braking isn't simply pads on a rotor, it depends on the friction between the driving surface and the wheel or tread. This is affected by gravity. We could get into the process of calculating the coefficient of friction, parallel motive force, blah, blah. The problem is that it depends on much more than just gravity. It depends on the material of your tires, the material of the street, the angle of ascent at any given moment, the square acreage of material against the road, etc., etc.. Suffice it to say, if you have good tire rubber, you need more of it on a low-G world than you do on a high-G world to compensate for the change in gravity. Unless you're planning to lift-and-lower wheels, there's not much you can do here but change tire widths between planets. But, one-size-don't-fit-all here on Earth, so you should expect that.
Suspension, on the other hand, is something you can do on the fly. If you were to design your suspension for the high-G world and try to use it on the low-G world, you'll find yourself bouncing all over the place. You could use a pneumatic suspension that adjusts the compressive force used for smooth driving by analyzing the weight (not mass) of a known mass (yeah, there's the mass). In other words, if your 1Kg mass weighs 4 pounds on your high-G world, the compressive force is increased by adding more air to the penumatic shock absorbers. That same mass may weigh only 1.8 pounds on your low-G world, thus air is removed.
Turning, is a form of braking — we just don't think of it that way, expecially when Hollywood has been advocating that a constant acceleration through a turn will increase your final velocity. That's actually true (of course, it's true without the turn, too. Hollywood tends to ignore that part.)— so long as your tires actually remain firmly fixed on the road. Exceed the force of friction and you slide off the cliff, usually in a brilliant (if unexplained) fireball. The flaming tire bouncing into the distance is mandatory. However, while your vehicle's onboard computer is calculating the amount of air to force into your pneumatic shocks (and your air brakes, for that matter...), it can also be calculating now much to lean the tires into the turn, ensuring greater friction on the lower-G planet.
But what about hovercraft?
OK! Hovercraft don't touch the road, but (today) use skirts to hold a cushion of air beneath the vehicle. You have a propeller for motive force, but otherwise the same rules apply to the engine (although the operator is manually increasing the throttle rather than the engine vacuum system doing it. No friction on the ground, no increase in torque.). In fact, other than having to increase the air pump system to hold the unit up on the higher-G world and a bit more throttle on the prop to climb a hill, there's not much difference in how hovercraft would be handled.
But what about real hovercraft? You know, the stuff we haven't invented yet?
OK, you're not holding that hovercraft above the ground on a cusion of air, your riding the magnetics (hard) or using anti-gravity (harder). In reality (if that word can be applied to something we can't actually do in reality), it's all the same problems. More force toward the center of the planet to hold the vehicle above the ground and more force behind the vehicle to move it up a hill.
So, one size really doesn't fit all
No, it doesn't. The design of a dump truck is very, very different compared to the design of a scooter. The fundamental physics are the same, but the needs and purpose of each vehicle vary so dramatically that it's impossible to claim that one size could or should fit all applications. And since you shouldn't do it in real life, you shouldn't do it in your story, either.
But, for the sake of argument, lets assume that your world is building a one-size-fits-all colonizing space ship. The Atlas Corp doesn't want to customize for every planet, they want to send the same thing to every planet (to manage costs). In that regard you might legitimately have a use for limited-design vehicles, in which case my suggestions apply.
But there would be so much customization anyway that I don't believe it's practical to think about it.
But, then again, I can believe that the tried-and-true bicycle wouldn't be different due to gravity. Two wheels, 1.5 inch wide tires, gears and brakes. The bicycle probably wouldn't change at all.