Can someone request phizer vs moderna at tinley convention center? Is there a private space available for someone on the Autism spectrum that may react to the situation?

No you are assigned a vaccine. The process is fairly quick

I was there mid-day on a Monday, and there was hardly anyone there. They told me weekends and typical "after work" hours are quite busy.

No. You are assigned a vaccine. Also, it is a wide open convention hall with lots of separated desks and areas to be directed to at each stage. I did not see a private space, so you might need to ask