Can getting the vaccine give you Covid?

OK, title is somewhat clickbait, but it kinda did happen to a friend of mine whom I had not seen since pre-pandemic days, and he mentioned that he had Covid last March.

He had no side effects from his first Pfizer shot, but started feeling bad a few days later and went to the Doctor. He was confirmed as having Covid, and while he was very sick for several days, he recovered without going to the hospital.
In getting info from him to submit to CDC, the Doctor told him it was almost certain that he acquired the virus during the 45 minutes or so that he sat in the Walgreens waiting area to get his shot. He said that the virus starts to grow as soon as you get the necessary load of viral spores, while the vaccine takes days to begin to offer protection, so he basically had to endure a non-vaccinated case.

He said it was a funny story, but funny only because he survived it.