No you can't. For one there isn't that much vaccine available. But the logistics are more difficult than that.
Gillette Stadium and Fenway are mass vaccination sites here in metro Boston, and Gillette will get up to 5,000 a day. There are staff on hand, and a waiting area post vaccine to handle the reactions. Most are none to minimal, some are not and some are severe. Meanwhile, you've got to distance everyone through all of that.
My mom and dad got the vaccine down in Beaufort, SC, and it was drive up. Meaning you never got out of the car. Stick your arm out the window for the shot. They did it on the local HS football field. You then parked for ten minutes in the waiting area and a nurse checked on you for adverse reactions before clearing you to leave. Distancing is completely solved in that situation. They are going faster than we are.