Yes, your idea is paranoid. Seek therapy because "the staff" at the place you went, or at any other healthcare facility don't inject patients with hiv.
I think what anxiousnomore is getting at is that there is no reason to assume that they are reusing needles or that someone with HIV was given the vaccine before you randomly. That's what we call anxious thinking. Usually when someone begins imagining these things, they have a history of anxiety. We can't answer that for you but it would strongly be suspected both from your question and your urgency and level of worry. Treatment for anxiety is available. The Us, Europe, in fact, most of the world follows protocols and those that don't make the news. There is no reason at all to suspect that the health agency that is providing vaccines to help people not get covid and not risk death are using used needles that might spread disease. Think about that. Okay?
Injecting with a used needle is a risk. But you know that. This didn't happen. The vaccine component question doesn't make sense and would impact nothing related to hiv. What would? Proper protocol which your country would be on the news if it didn't follow. So, please talk to someone about anxiety and see if you can not have these fears so easily.