Yes! The Pfizer vaccine, for example, enrolled 45,000+ people, of whom 25,000+ people got the actual vaccine. This is on par with all other non-COVID vaccine trials, who usually enroll 30,000+ people.
Also keep in mind that other medicines (pills) are approved after only being tested in much fewer people (~1500 people, for example. Vaccines are the most rigorously tested therapies, including the COVID vaccine.