If you want the real thing you're going to have to invest. SnoWizard of New Orleans makes home machines, but they are not cheap; it will set you back about the price of a long weekend onsite at WDW.
http://snowizard.com/shavers/cube/
(I think what I'd do first is call SnoWizard and see if they have any business customers in your area; you might just have not found the stand.) Personally, in your shoes I'd be tempted to go with a pro machine and set your kids up in a garden shed out front -- they can pay for college.
(Pro machines are darn near indestructible, btw; I know people in Louisiana who have been using the same one daily seven months out of the year for fifty years; all you have to do is replace the motor and belt once in a while. You can find used machines in the New Orleans area pretty easily, but those are hard to come by elsewhere. I wasn't kidding about college; my best friend and I used to work her family's stand -- we started in second grade, standing on a crate.)
Snowballs are an odd thing, and you'll only find them in certain places, usually places that have a lot of connections to New Orleans (though the Hawaiians have a similar tradition that they call a shaved ice, and their machines are differently made.) We do have them locally; it is one of the things that makes living in this (midwestern) city bearable in summertime. I hear that they are very popular around Baltimore, too.
I've noticed that in Florida good snowballs do tend to be kind of hard to find except in the Destin-Pensacola corridor. Ponte Vedra may well zone against them; my guess is that you'd have better luck finding a stand in an older neighborhood in Jacksonville. ...
YEP -- I looked. There is apparently a stand that operates at the corner of Hartley and St. Augustine Hwy. in Jacksonville; it's called Snoballs. There are apparently two others at the Beaches that use the Hawaiian-style machines, one in Atlantic Beach and one in Jacksonville Beach.)