If you have a laptop, then it is via dial-up. The service would be whatever service you subscribe to. Hopefully it would have a local or toll-free dialup number, so the meter is not ticking when you are connected.
I've taken my laptop to BWV, VB, and HH and had no major problems. It's best if you get the dialup numbers and protocols established at home ahead of time, while you can easily get online to look them up. If you get to Orlando and don't know the dialup numbers or configurations, it takes a bit of effort to find out by voice telephone.
I don't know of any Internet Cafe services at Disney, but there may well be some by now.