I've been trying to find a way to measure the speed of a car while in/on the vehicle using some sort of speedometer but can't seem to find anything that doesn't rely on measuring the rotation of the wheel. Are there any speedometers that I can use that can just point towards the ground or something and get a speed reading on the fly?

you always need one point (or more) as reference,
If you have a PC with a high speed camera you can compare the offset of 2 images to determine the speed.

In the Netherlands there are distance signs every 100 meter near most main roads. You can create a stopwatch that is pressed when passing 2 of them.

A nice speedometer could be measuring the wind speed on top of your car. Measure it when you start and measure it while driving.
You need a compass to compensate for your start speed depending on the angle you drive - some sin() and cos() math comes in.

Another idea I can think of is to have a microphone measure the sound of your car and do an FFT to determine a signature of the speed. Don’t know if it works but I know my sounds makes different noises depending on its speed.

This last one leads me to the following idea: you could try to make a set of resonators that each have their own “resonation speed”.

my 2 cents

What about something like this: http://arduino.cc/forum/index.php/topic,24337.0.html

Using a GPS unit would give you the most accurate speed.