Does a 4x8 sheet of 1/4'' pegboard need support across the middle of the board?
The board should hold the weight you describe vertically. The board will probably bow a little bit naturally even without tools on it.  I would add the extra stud because it's cheap, easy, and better to do things right the first time.

I think 2" x 4" studs are a bit over-kill here.  You could probably use 2" x 2" boards to make it lighter (easier to mount) and not stick out from the wall so much.

Pre-drill all your screw holes.  If you try and screw into it near the edges without pre-drilling, the board could tear.  That sounds bad; it really isn't, just looks noticeable.