The control group is the untreated population outside of the study. I'm not just speaking figuratively. The standard experimental design is to state the null hypothesis (i.e. the vaccine does not lower infection rates), and look for evidence to reject it. That evidence would be a statistically-significant rate of infection that's lower than the untreated population. Note that this explicitly defines a control group.

Really, too much is made of the placebo effect, such that it frequently turns into magical thinking, that a person can avoid illness with a healthy set of beliefs. "I don't believe the medical quacks who say I 'snapped my femur.' I'm going for a walk!" Nonsense. The double-blind RCT is good for, e.g., drugs that treat symptoms, the experience of which is subjective. It's not needed for objectively measurable conditions. Infectious disease agents have no respect for state of mind, and you get the disease whether you believe in it or not, as we saw during the pandemic.