There are free public access computers in several libraries in that area.
The library that's in the same strip mall as the Publix near Orange Lake Country Club and Resort has at least 8 of them. There's rarely much of a wait but you are limited to 30 minutes at a time, up to twice per day. The system locks you at at precisely 30 minutes so be sure to save whatever you need onto a flash drive if doing school work.
There's another small library branch just off Rte 192 not far from the Celebration watertower. From 192, turn into Blake Blvd, as if you were going to Joe's Crab Shack Restaurant, but look left, at the shops along that strip. There's a tiny storefront library with 4-5 computers. This one almost always has a long wait to use a PC because they let you use it for 60 minutes at a time, up to twice per day. If no one is waiting (rare - maybe EARLY in the AM), you can stay on longer. They just tell you when it's time to let the next person, so you have a chance to save your work.
The other counties in the area don't have free computer access, they charge a weekly or annual fee and they still limit usage to 30 or 60 minutes per session.
We're spoiled by our stays at Wyndham (previously Fairfield) timeshares. Most have at least one computer near the office for registered guests to use for free - we just sign up for a time slot and show up. HTH!