For naval officers, it's expected to get the Master's degree sometime before the promotion board for the rank of Commander (that's the one after Ensign, Lieutenant junior grade, Lieutenant, and Lieutenant Commander) at the latest. As noted above, a tiny number go right after USNA (IGEP, or Immediate Grad Ed Program), but routinely, everyone else heads to the Fleet (or Corps) to do what junior officers are expected to do. Be young warriors!
There is also the VGEP program, for, again, a small number of mids, who are ahead enough on their matrix and with the grades to qualify, to finish USNA undergrad work by December of senior year and take grad courses at a local university (Georgetown, American, Maryland, GW, Johns Hopkins, etc.). They graduate with their USNA class but stick around in student mode to finish up the Master's by December, then joining their classmates in the Fleet or training pipelines.
Later on, there are a number of programs, Naval Postgraduate School, CNO Intern, Naval War College both resident and non-resident, Tuition Aid (going to school at night using Navy money) and many more. If you are doing well as an officer and tracking for promotion, you will be given opportunities to get the Master's degree - regardless of commissioning source.
As for an SA background providing a better shot, I would say it's not a critical element. I think VGEP is unique to USNA, there may be some more opportunity to attend a marquee school, and certainly the undergraduate preparation is stellar, so that acceptance to grad school later on shouldn't be a problem.
No matter what, midshipmen get plenty of briefs at USNA, as I suspect other SAs do, and hear about other programs along the way after they are commissioned. The first shore duty after being at sea is the usual first window for a Master's.
Search around for various threads on the graduate school issue.