Hi Suzanne,
Thank you so much for visiting the Disney Parks Moms Panel with your question. A trip to
Disney's Vero Beach Resort sounds like so much fun. Relaxing on the beach at a Disney Resort sounds like a lovely vacation!
There are a couple of dining options at Disney's Vero Beach Resort, including
Wind & Waves Grill. If you are able to with your schedule, do consider checking out Goofy's Beachside Character Breakfast at this location. It is offered on Saturday mornings.
As to locations in the off-site area, some options include Squid Lips Overwater Grill, Ay Jalisco, Orchid Island Pizzeria, Ocean Grill, and Mr. Manatees. While some of these are close enough to be in walking distance, others will require a drive to get to them. You can also always stop at the front desk to ask a cast member for dining recommendations as well.
I would like to thank my fellow Disney Parks Moms Panelists, including my fellow Disney Vacation Club specialist Trish, who helped with recommendations for this answer. Please do let us know if you have any further questions, Suzanne!