If so, couldn't it also rotate itself sideways somewhat by turning its front wheels? (assuming it has 4wd and electricity.)
Tl;dr: Yes, it could. You'd have a little pitch authority and even less yaw authority (defining pitch axis as through the doors and yaw axis as "up" in the sense of a car on Earth).
Assuming, of course, that it was set up to be controlled from the ground, has some sort of power source, and none of the engine components have cold-welded (this is a big one: https://en.wikipedia.org/wiki/Cold_welding) or outgassed severely enough that they no long function. Machines that weren't built specifically to operate in a microgravity, near-vacuum environment generally don't.
Spinning up the wheels would induce a pitching moment in the opposite direction, per conservation of angular momentum. However, the wheels are so light compared to the mass of the full vehicle that the rotation would be VERY slow. Reaction wheels are generally designed to be as massive as allowable by your mass-budget: increasing either the moment of inertia of the wheels (by increasing the mass, assuming the shape remains the same) or their speed will increase the maximum torque that can be applied.
Turning the steering wheel would also change the angular momentum of the wheels (in direction, not magnitude), but this effect would be even smaller than that of the initial spin-up (but would be a slight yaw instead of pitch).
Assuming that the Roadsters center-of-mass is not perfectly in-line with it's center-of-pressure, there is an induced torque from solar radiation pressure (SRP). If the Roadster were oriented such that this torque produces a pitching-moment, it could theoretically resist it by spinning the wheels to produce an opposing moment. However, it would need to keep spinning the wheels faster and faster to resist this moment until eventually it reaches the hardware limit of the wheels. This is called reaction wheel saturation, and requires another form of attitude control (usually RCS thrusters in deep space) to hold the vehicle steady while it lets the wheels spin down (this is called "desaturation")*.
*Interesting aside: depending on the spacecraft orientation and the nature of the disturbing torque, sometimes the wheels actually slow down (alternatively: speed up in the opposite direction) to resist the disturbing torque. It is often undesirable to have your wheel speeds near zero due to increased/not-well-behaved friction at those speeds (see "sub-EHD": http://esmats.eu/esmatspapers/pastpapers/pdfs/2017/bialke.pdf), so the wheels are actually "resaturated" as opposed to "desaturated". That's all just a matter of what convention you use when defining angular momentum, though.
Would Elon Musk's roadster be able to rotate itself using its wheels as reaction control?
If it had power and if the wheels were free to move, yes.
However, it's worth noting that once you start using the wheels for reaction control, you can't turn them off without either dumping momentum first or causing a strong counter-reaction.
The specific axes available for control would depend on the orientation of the wheels (which could be changed a little with steering).
It would have been cool to include a set of small cold-gas thrusters and some solar cells with the roadster. Oh well.