Hi mez9757, welcome to CruiseReviews! Congratulations on your upcoming cruise! I did the Jeep Safari, many, many years ago. It was a lot of fun and you got to see a lot of the island. They do drive on the opposite side of the road there. Since St. Thomas is an easy island to do on your own, I am more inclined to say you'd be ok to wait. However, if this is something you really, really want to do and would be disappointed if you couldn't, then go ahead a book it on board. What few dollars you might save would be made up by peace of mind. And the fact that if you're on a ship's excursion and something goes wrong, the ship will wait for you. If you're not on a ship's sponsored excursion, the ship will not wait. Since this is your first cruise, I'd hate to have it tainted by a misfortune.
Most of the tour operators are very efficient and will have you back in time. However, the traffic on St. Thomas when ships are in port is horrendous. That plus there's always the possibility of a flat tire or the jeep breaking down. Not that it happens often, but there is the possibility.
Just my two cents
I hope you will have perfect weather, clam seas, and a marvelous cruise! Just be warned, cruising is addictive