I'm happy in any kind of aircraft, including flexwing microlights, which are open cockpit and very small. I was even OK when I tried hang gliding. But I've always been scared of heights, and still have difficulty climbing ladders. I used to be scared in airliners, but I'm fine now I understand what's going on. The only time I've got vertigo when flying is at low level in a helicopter in the South Wales valleys; at one point the ground dropped away quite suddenly, and that gave me the sensation of height that you get on a cliff top or similar. But in a small aeroplane you don't fly that low, so it won't happen.
So as everyone else has said, give it a go. Go for a trial lesson, or get someone to take you up. It might be worth mentioning your fears though, so the pilot or instructor knows you're nervous; don't worry, you won't be the first.