Android - How can I prevent my screen from turning off while on a phone call?

Keep screen on during calls

That setting is not available on all phones, but if it was it would be under: Home > Settings > Call Settings > [Display >] ...

But as for the difficulty turning the screen on

Be sure that the power button does not end calls (Settings > Accessibility > Power Button ends call : make sure it is UNchecked)

Then just pull the phone away and press the power button to turn the screen on.

Are you sure that the screen goes off after X seconds? Android phones with proximity sensors usually turn off the screen when they are put on an ear. The Phone should automatically turn on the screen if you uncover the light/proximity sensor (e.g. by removing it from your ear). The Galaxy S does this and I suppose that the Vibrant does so too, because they are basically the same phone models. Other Android phones behave similar.

I am not sure if it's possible to disable this feature. Even if so, you would encounter another problem: Accidental pushing buttons on the touchscreen with parts of your face, which could lead to hang-ups.

I had been having exact same problem for a few months on my LG-P500, but have just fixed it. I found that the proximity sensor had gotten covered with dust as the screen cover had slightly lifted in the area of the proximity sensor. Once I cleaned it off it worked fine again.