Are you automatically disqualified for acceptance of any branch of the military having missing a finger?
No, though you will have to apply for, and be granted, a Medical
Waiver by the US Military's Medical Corps. Basically, this waiver
states that your missing digit(s) do not significantly hamper your
ability to serve effectively. You would be unlikely to be granted a
waiver if you are missing a thumb or index finger of either hand,
or missing more than one finger on a hand.