Create a polygon cylinder (radius = 1.0; height = 0.3; subDivs around axis = 20; subdivs on caps = 0; axis = z;), rename it to "wheelFL" - this will be our front-left wheel. Delete its history.
Scale the wheel by 2.0 in x-direction. Create a locator "locFL", make it a child of the wheel. This will be the contact point with the main body later on.
Note: Of course you are free to use a nurbsCylinder, too.

Dublicate the wheel 3 times. Rename the new wheels to "wheelFR", "wheelBL", "wheelBR". Update the child locators accordingly, too.
Move "wheelFL" to [3,1,-3], "wheelFR" to [3,1,3], "wheelBL" to [-3,1,-3], "wheelBR" to [-3,1,3].
Create a polyCube "body" (w=10; h=1;d=5). Create a locator "vehicle", move it above the vehicle parts and freeze its transforms.

First, parent all vehicle parts to the "vehicle" locator. We will use this locator to move the vehicle later on.
Your outliner should look like this:

We will now turn the "vehicle" locator into the interface to the expressions, driving the animation of our vehicle. Add custom attributes as shown on the left to the "vehicle" node.
The ones ending with "_EXPR" are of type "integer" and their purpose is to give us easy access to the respective expression.
The other ones will serve us as a means to enter a rotational offset for each wheel. Ther are of type "float".

// helper connection to the interface attribute
vehicle.wheelFL_EXPR =0;

// constant values
$pi = 3.141592;

// radii of the elliptical wheel
$rA = 2.0;
$rB = 1.0;

// get translation of the vehicle and offset of the wheel
$x = vehicle.translateX;
$rotZOffset = vehicle.rotOffsetFL;

// as we are working in radians, convert the rot offset
$rotZOffset = deg_to_rad($rotZOffset);

// compute perimeter of elliptical wheel
$per = $pi*($rA+$rB);

// compute the rotation of the wheel
$rot = (-2*$pi*$x/$per)+$rotZOffset;

// adjust the y-position of the wheels center
$y = abs((-sin($rot)*tan($rot)*$rA*$rA)-(cos($rot)*$rB*$rB))/
     sqrt($rA*$rA*pow(tan($rot),2)+$rB*$rB);

// apply the results to our wheel
wheelFL.translateY = $y;
wheelFL.rotateZ = rad_to_deg($rot);

// helper connection to the interface attribute
vehicle.body_EXPR =0.0;

// get positions of contact points of every wheel
float $pos[];

$pos = `xform -q -ws -t locFL`;
vector $posFL = <<$pos[0],$pos[1],$pos[2]>>;

$pos = `xform -q -ws -t locFR`;
vector $posFR = <<$pos[0],$pos[1],$pos[2]>>;

$pos = `xform -q -ws -t locBL`;
vector $posBL = <<$pos[0],$pos[1],$pos[2]>>;

$pos = `xform -q -ws -t locBR`;
vector $posBR = <<$pos[0],$pos[1],$pos[2]>>;

// compute mean position
vector $meanPos = ($posFL + $posFR + $posBL + $posBR) / 4.0;

// [..]

// [..]

// get x-angle between the left and right contact position and
// the z-axis, do this for front and back, our rotation angle
// will be the mean angle

float $angleXF = `angle <<0.0,0.0,1.0>> ( $posFL - $posFR )`;
if ($posFL.y > $posFR.y) $angleXF = -$angleXF;

float $angleXB = `angle <<0.0,0.0,1.0>> ( $posBL - $posBR )`;
if ($posBL.y > $posBR.y) $angleXB = -$angleXB;

float $angleX = ($angleXB+$angleXF)/2.0;

// get z-angle between the front and back contact position and
// the x-axis, do this for left and right, our rotation angle
// will be the mean angle

float $angleZL = `angle <<1.0,0.0,0.0>> ( $posFL - $posBL )`;
if ($posFL.y < $posBL.y) $angleZL = -$angleZL;

float $angleZR = `angle <<1.0,0.0,0.0>> ( $posFR - $posBR )`;
if ($posFR.y < $posBR.y) $angleZR = -$angleZR;

float $angleZ = ($angleZL+$angleZR)/2.0;

// [..]

// [..]

// apply results to body, take into account that "body" is
// parented to "vehicle"

body.translateX = $meanPos.x - vehicle.translateX;
body.translateY = $meanPos.y - vehicle.translateY;
body.translateZ = $meanPos.z - vehicle.translateZ;

body.rotateX = rad_to_deg($angleX);
body.rotateZ = rad_to_deg($angleZ);

Here you are!

Some suggestions:
- try different rotational offset settings
- change the size of the wheels (don't forget to adjust the radii in the expressions!)
- move the position of the contact locators on the wheels