Just played the New Course Sunday @ Grand Cyprus Villas. It's a links style course and it was pretty good and in great shape. Gotta pay attention to the starter when he tells you which flag to shoot at on the double greens.
There are two other courses within Grand Cyprus Villas and it's not too far from Disney Springs.
Magnolia and Palm are the courses to play @ Disney and they're near Magic Kingdom, near Grand Floridian and the Polynesian. Not a fan of Lake Buena Vista course which is near Disney Springs/Saratoga Springs.
If you want to splurge a bit, you can try to get on @ Tranquilo Bay Golf Club @ Four Seasons and it's actually within Disney property. Never played it so not sure if you can get tee time without being a resort guest.
If you have a car and willing to drive a bit, you can play @ Orange County National Golf Center. There are 2 18 hole courses and a 9 hole executive course. Facilities and the course are in good condition and they host Q-School event at the facilities every year there. About 15 minute drive if you don't get lost within Disney.
Of course there is always Bay Hill, about 30 minutes from Disney.