lol, pepper

dizziness and feinting were the reactions, not really uncommon with people afraid of needles
Those reports are a bit vague--are those people actually fainting? Fainting involves losing consciousness; "feeling faint" or "feeling lightheaded" is quite common in people who fear needles or may be just very anxious.
Having said that, every Covid vaccine (Pfizer, Moderna, AstraZeneca, and J&J) are administered by injection. I have only seen reports about fainting with J&J, but it may be a fluke. Other vaccines do cause, sometimes, dizziness, lightheadedness and sweating, and even anaphylaxis (very rare but also potentially lethal if not treated immediately), but I haven't seen any reports of actual fainting. Again, doesn't mean it doesn't happen.
ETA: And now we're seeing first reports about blood clots associated with J&J (not unlike with AstraZeneca, which is actually similar to J&J ). Again, it's still extremely rare and may not be at all related to the vaccine, who knows. But since the blood clots apparently appear 3-4 weeks after the injection, everybody is watching.