I have noticed a very strong vibration that occurs randomly while driving. It vanishes when I stop accelerating but if I start again it reappears. It also only tends to appear around 2500 rpms and around 45-55mph. The vibration seems to come from the bottom of the car in the center of the driver and passenger would sit.

 

Sounds like a driveshaft issue. I don't think its anything else because you said the issue goes away when you lift off the gas.
Here are a few things it should be:

-u-joints (when these become dry or when they fail, they tend to produce a bad vibration around that speed.
-Center support bearing (when this becomes worn, the driveshaft will bounce around at a certain speed)

crawl under there and grab the driveshaft and move it around (back and forth, and also clockwise/counterclockwise) if you hear clicking or feel excessive play, then I would look into replacing the u-joints and the center support bearing. both inexpensive parts.


edit: all the above assuming you have an AWD.