Vaccine deals are already negotiated at the Government levels. Vaccines are in short supply and the suppliers cannot meet the already committed demand.
The USA managed to increase their production by improving the manufacturing techniques, currently scheduled to make 4.7 billion doses by the end of the year.
In the chain there are a lot of input materials needed, so you just cannot accelerate as you wish. On the manufacturing side, Bio reactors, pumps, Filtration units, cellular materials, adjuvants, lipids - then onto distribution filling up glass vials, refrigeration - then finally the syringes and the needles. The list isn't complete here but all the major ones are. It's no easy task to handle when it's needed by the whole world.
Once USA finishes the current vaccine drive, there will be some vaccines available for private sales. But they are going to establish a governing body to handle the future distributions, which is a fair call.