Melbourne, FL is actually the closest airport to Vero, about 45-60 mins away; also the closest to Port Canaveral. It is a lovely little airport, easy to navigate and friendly, but means a change of planes for most folks. We generally prefer to fly into MCO so we can get a direct flight, but if you can get a good deal to MLB or you have to change planes somewhere anyway, I'd recommend MLB.
MLB is 24 miles from Port Canaveral, and 34 miles to Disney's Vero Beach Resort (which is geographically in Wabasso Beach).
From the resort to the port, it might take 80 mins (all on 1A1 along the barrier islands between the ocean and the Indian River, so pretty views, but maybe some congestion in settled portions).