|Why does the steering wheel vibrate?|
As the steering wheel is connected to the wheels, the problem lies in the wheels and engine only. If the wheels are not aligned properly or if your wheels have slipped belt i.e the wear and tear is only in a specific zone then your wheels tend to wobble. This in turn makes the steering wheel to vibrate.
Another reason can be resonance. Earlier car bodies and their suspensions were not adjusted to high speed limits. So when you accelerate your car to higher speeds then the whole car body and suspension starts to vibrate. This may stop when you cross the range of resonance speed at which it vibrates.
Even if the engine is weak and not powerful enough to push your car to higher speeds, your steering wheel vibrates due to the internal vibration of the engine. The reason is that the engine is not able to bear the load and is not able to pull the car to higher speed due to lack of power. This problem is dealt in modern cars by implementing heavy dampers and shock absorbers.
There are several other reasons like bent rim and sometimes the joint of the control valve is not properly connected.
So even if your car is old, don't worry. Just realign your wheels and fasten some bolts to get some relief at higher speeds.