For direct flights from BWI, you are probably looking at PBI (West Palm) or MCO (Orlando). Melbourne is definately the closest, but it's a small airport, and there are no direct flights there from BWI. There is tons of direct service from BWI to MCO, and Southwest flies BWI to PBI.
MCO and PBI are about equidistant from Disney's Vero Beach resort, tho PBI is a much easier ride, being a straight shot down I-95 (PBI now has its own offramp from 95). MCO takes only a little longer, and there's more competition there, so you're likely to get better prices.
Good luck!