In general at any of the parks when thunder or lightening is 'in the area' all outside activities (especially water activities) are halted and stay stopped until there has been no sighting / detection for 30 minutes. This is why it is a good idea to get the dolphin swim in early as afternoon thunderstorms are pretty common in the summer. (This is what the weather service recommends ... I worked at an 'attraction' and they came once a year for training and this is what they said and management followed . . .)
As for a 'storm' this is what DC has in their FAQ:
In the event a named tropical storm or hurricane approaches Orlando, Tampa, or a guest's hometown, the parks will reschedule or refund any vacation package and/or individual park tickets booked through SeaWorld Vacations, Busch Gardens Vacations, DiscoveryCove.com, SeaWorld.com, Aquatica.com, BuschGardens.com, or the call center. The parks will not apply any cancellation or change fees for this service. For additional Discovery Cove information call 407-513-4600.
