Cellular service is available through Cellular at Sea providing service for the major carriers but is extremely expensive. You would have to add international service to your plan to avoid paying even higher rates (even with international plan the rates are relatively high). Verify cost with your service provider prior to departure. The best way to contact anyone is through the Wave phones provided by Disney for the duration of the cruise and work both throughout the ship and on Castaway Cay for both voice and texting. These do not work for contacting anyone or anywhere outside the cruise you are on.
There are no Wi-Fi hotspots on Castaway Cay. Wi-Fi is available on the ship for an additional cost.