Most modern courgettes are bush varieties which stay in a fairly compact area compared to the old trailing ones. In either case it IS possible to tie the leading shoot into a vertical support like a cane tripod to achieve a 'vertical' plant, but in my experience you have to be very vigilant to keep pinching out sideshoots. You will certainly get a bigger crop by just leaving it to do what it wants. You can pinch out unwanted sideshoots to concentrate the growth in two or three directions.

I grew one in the greenhouse last year that I was a bit frightened of, it got so big! Hundreds of courgettes though - it went on for months. I have put on in again, but beside the door this time, rather than the other end of the greenhouse and I am going to try growing it vertically.