More than you would think. Most people don't show up to ask to do research in my opinion. I got my first one by literally walking up to a Prof after class and saying "I like this. Do you know anyone who does research like this?". He said "me", gave me a test project, and I worked for him for like almost 20 years.

I should say I also did this my freshman year (the year before the story above) and I got referred to a different guy in the department where I worked for a summer but it didn't work out for either of us. Turns out I'm more engineer than physicist, but it was good knowledge.

Edit: To be more direct about numbers, you're right there's not a spot for everybody or even half, but when I was in undergrad way less than that even went looking for research, they just did the job fair every year mostly and were students the rest of the time which was... not as reliable a strategy we will say.