Hi Camille,
Without a car, your two options are taxicab or public transport. The bus will be cheaper than the cab! Take the Disney Resort buses to the TTC (Transportation & Ticket Center). From here you need a Lynx 50 heading toward Downtown Orlando (return journey terminates at Magic Kingdom).
It's about 25 minutes to Seaworld, the bus stops on the street outside the parking lot. $2 each way, or if you plan on doing a lot more bus exploring on a day, $4.50 buys a one-day go-anywhere pass for the Lynx network.
The route/schedule for the bus is available here:
http://www.golynx.com/assets/userfiles/routemaps/50.pdf
Hope that helps!