Nearest one was about 8 miles last time I was there. Super Wal-Mart north at Sebastian and Publix in VB were the 2 best options. I believe I heard there was a new one but am not certain.
Cross the bridge just at the light just north of the resort entrance (510 West) and go to US 1 and turn right. Go north a few miles and a new Publix is on the left. The older Publix is about 8 miles north of 510 in Sebastian (near the Walmart). The new Publix is about 6-7 miles from the resort.