All the venison, 'wild boar,' bison, ostrich, rabbit, etc sold in restaurants (in WDW at least, this isn't an Adirondack lodge or a real life road kill cafe, after all) are farm raised. 'Wild boar' is not actually wild, it's a breed - which is too bad because there are areas in the south that actually have an invasive wild boar problem when boar brought over to breed with pigs got loose, they're destroying lots of woodland. It would be nice if they were rounded up and served to adventurous restaurant patrons just to get rid of them.
So, in short OP, can't help you. There's no wild game in WDW. Not even the occasional alligator or rattlesnake seen on Florida menus is wild, it comes from farms. But if you're staying at Fort Wildnerness and you're very light on your feet there's possum, squirrel, and lots of little lizards you could grill up on the conveniently provided grills (I've been informed the only grills at WDW resorts are in Fort Wilderness). I believe the fish are officially off limits. I'd bring a doctor's note explaining your condition, in case anyone gets concerned when you run by with a net.