Absolutely possible and not terribly difficult to pick up a rental at port. Though that may depend on rental agencies in Vero Beach.
When you get off the ship, you'll be directed to the car rental shuttles. Get on the right one and head to the agency to get your car. I do suggest having a reservation in advance as they get very busy. There are many car rental agencies with locations in Cocoa Beach right near the port; check back frequently for deals. Most will probably charge a one-way fee (maybe not to MCO but likely to Vero Beach) so make sure you know whether and how much fee is involved.
Enjoy your cruise!