I literally thought of that example last week when the Pfizer vaccine came out. I got some Omaha steaks a few months ago, and it came with the CO2 - no doubt that CO2 will be shifted to the vaccine if it's approved and Moderna has some kind of problems.
I imagine there will be common bottleneck points, too. Even if Moderna can make 10 million doses, and Pfizer can make another 10 million doses of their vaccine, that doesn't mean that 20 million doses (for 10 million people) can be distributed. The vaccines have supplies in common that may be limited in quantity. You need vials, syringes, alcohol pads, gloves and other PPE, and most importantly, people to give the vaccine. I think it'll be awhile before it's widely available.
I got my flu shot in my car - they were pretty quick, and had two stations set up to deal with 12 people per 10 minute time slot. Hopefully local departments of health can do something similar.