I live on South Padre Island although I am in Denver right now. I just bought a great 2005 Jeep Wrangler with 19,000 miles. I have wanted one for ages so I could navigate the beach all the way to the jetty at Port Mansfield.
That being said, I hope you enjoy your time on the beach but please do not go 4 wheeling in the dunes. SPI is a barrier island with a fragile eco-system. The dunes need to be protected from excessive disturbance to develop vegetation, discourage erosion and ultimately provide protection to the mainland from a catastrophic storm surge.
We are very lucky to have the ability to drive on our beaches and you can find some unbelievably solitary areas on the "north beach" from which to enjoy the sun, gulf, hunt shells or sea beans, surf fish, and generally have a great beach experience.
Come to the island, take your jeep north, only please respect the dunes. I love going on trails in the mountains and will try any back road -- but the dunes on SPI are really too vulnerable to be used for off-road adventures.
Have a great Memorial Day on SPI!!