No. The AMA does not have any control over how many students are accepted to medical school. The AAMC does, and there technically is a cap, but they've been intentionally increasing it steadily for over a decade in order to increase the supply of doctors.
However, even that's irrelevant, because the number of people graduating medical school isn't a bottleneck for the supply of physicians - residency programs are. There are far more doctors than there are residency slots, and the AAMC doesn't control residency positions. So whatever cap they have at this point doesn't affect anything other than the number of people who graduate medical school and end up having to get a job in a different field.
It's the other way around. Resdiency positions are extraordinary expensive to offer - remember that this is where doctors actually receive their clinical training. Medicare funds them, because otherwise hospitals would have no incentive to offer them (they would lose money on each resident).